Transmission zeros created when selecting outputs in state-space system

I have a MIMO state-space system with inputs of rotor speed (quad-rotor plant) and outputs of the states (U, V, W, Phi, Theta, P, Q, R). This system does not have any transmission zeros. As I create a smaller plant with the same inputs and a reduced output set (W, Phi, Theta, R), two transmission zeros are created. I arrive at the same result when using balred() and minreal() to reduce the plant order.

Is it possible for a plant to create these transmission zeros or is this an artifact of how MATLAB reduces the plant model? Can I trust that these transmission zeros are real?

Thank you for your time, Clay

[Martin Evans]

I have exactly the same issue and I haven't been able to work it out. I can only guess it's an artifact unintentionally added in the balred function.

Answer 1

[Shreyansh]

Let's walk through it carefully.

First, yes, it is absolutely possible that transmission zeros appear when you reduce the number of outputs (even without changing the inputs). Here's why:

• Transmission zeros are related to the ability of inputs to affect outputs through the system dynamics.
• When you select a subset of outputs, the "pathways" between inputs and outputs change. In the full system, certain cancellations may not occur because of other outputs balancing things out.
• When you drop some outputs, you can "uncover" cancellations (zeros) that were not obvious or did not affect the full system.
• So, the new system (inputs → fewer outputs) is a different transfer matrix, and it can naturally have different transmission zeros, including new ones.

Now regarding your MATLAB reductions (balred(), minreal()):

• balred() tries to reduce system order based on Hankel singular values — removing states that are "less controllable/observable."
• minreal() cancels pole-zero pairs that are exactly (or nearly) identical.

Neither method fabricates transmission zeros artificially. However, both can sometimes slightly move system dynamics due to small numerical tolerances, especially in lightly damped or nearly uncontrollable modes.

But based on what you described — you first selected outputs, and then used balred() and minreal() — it's very likely that those new transmission zeros are real and reflect the changed input-output structure.

[Paul]

To illustrate with a simple example:

Mtf = [tf([1 1],[1 4 1]);1]

Mtf =
 
  From input to output...
           s + 1
   1:  -------------
       s^2 + 4 s + 1
 
   2:  1
 
Continuous-time transfer function.

Mss = ss(Mtf);

Mss doesn't have any transmission zeros

tzero(Mss)

ans =

  0x1 empty double column vector

But selecting only the first output does result in a transmission zero

tzero(Mss(1,1))

ans = -1.0000