# Zeros of a Trransfer Function

I would be grateful if someone could explain me what does the value of zeros of a transfer function tell about the dynamics of the system.

• First what does your transfer function represent ? output vs input ? – Sam Farjamirad Sep 17 '18 at 20:49

The zeros of a transfer function influence the dynamics of the system in multiple ways.

1. Consider the output of a given system for $u(t) = 0$ as a weighted linear combination $$y_a(t) = \sum_{i=1}^n R_i e^{P_it}$$ where $P_i$ are the poles of the system. The weights $R_i$ are influenced by the zeros.
2. For a zero at $s = N_i$, a signal with the complex frequency $N_i$ is not transmitted anymore, i.e. input signals $u(t) = e^{N_i t}$ are "blocked" by the system.
3. Zeros are invariant towards state feedback, i.e., while the poles of a system can easily be moved by a corresponding state feedback, that is not possible for the zeros.
4. It can be shown that a system with a zero in the right half-plane will always be unstable for high feedback gains and that the possible control performance for such a system is limited.
5. The zeros of the transfer function of the open loop describe an internal dynamic of the system, which is not observable. This internal dynamic was initially described by Isidori as zero dynamics.

The zero dynamics describe the internal dynamic of a system for the special case of an input $u(t)$ and an initial condition $x(0)$ such that the output $y(t) \equiv 0$ for all times $t \geq t_0$.

For further reading on this pretty interesting topic, I'd start right at the source with Isidori's book "Nonlinear Control Systems: An introduction"

• Nothing in the answer is specific to nonlinear systems, so why recommend a book on nonlinear systems? (Especially since the publisher's price is over £100) – alephzero Sep 18 '18 at 0:32
• @alephzero I only wrote it as a bit of a side note in 5., the book was recommended for the fact that it was the first to describe zero dynamics (can be applied to linear and nonlinear systems), not for its title. – OpticalResonator Sep 18 '18 at 5:26