# Pole placement with damping ratio and natural frequency

how can I find out the poles of a system if I have the damping ratio $$ζ = 0.707$$ and the natural frequency $$ω_n = 2.3\text{ rad/sec}$$?

$$A = \begin{bmatrix} -0.02 & 0.1 & 0 & -9.81 \\ -0.23 & -0.134 & 120 & 0 \\ 0.0006645 & -0.019598 & -0.97 & 0\\ 0 & 0 & 1 & 0 \end{bmatrix}$$

$$B = \begin{bmatrix} 0.14 \\ -2.9 \\ -0.6313 \\ 0 \end{bmatrix}$$

$$C = \begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix}$$

$$D = 0$$

• poles of the system are equal to the eigenvalues of $A$. As this is a fourth order model, the standard second order model notation is no longer valid. Is this the only natural frequency? These kind of models could have upto 2 resonance frequencies. May 11 at 14:33
• Also be aware that zeros cause overshoot (or even undershoot) as well. May 11 at 21:25

The natural frequency $$\omega_n$$ and damping ratio $$\zeta$$ of a system are computed by using the two dominant poles of the system. Dominant poles are the ones that are closer to the imaginary axis (i.e. their real part is closer to zero) and define the dynamic behaviour of the system. Since your system is of order $$4$$ the characteristic polynomial (denominator of transfer function) can be written in the following form:

\begin{align} P(s) &= (s+p_1) \ \cdot \ (s+p_2)\ \cdot \ (s+p_3) \ \cdot \ (s+p_4) \\ &= (s^2 + 2\zeta \omega_n s + \omega_n^2) \ \ \cdot \ (s+p_3) \ \cdot \ (s+p_4) \end{align}

where $$p_1,p_2,p_3,p_4$$ are the poles of your system. In order to find the poles of you system you can either transfer your system's representation into the s-domain (i.e. obtain the transfer function from your state-space representation) and compute the roots of the characteristic polynomial (denominator of transfer function) or you can directly compute the eigenvalues $$λ_1,λ_2,λ_3,λ_4$$ (your system has $$4$$ eigenvalues since it is of order $$4$$) by using the following determinant:

$$det(\lambda I - A) = 0$$

where $$I$$ is the identity matrix. The solutions of this equation (they will be $$4$$ values for $$\lambda$$) are the eigenvalues of the system which in your particular case will also be the poles of your system (this is not always true and almost everyone confuses it).

Of course, if you want to find the poles of your system if the damping ratio $$\zeta = 0.707$$ and the natural frequency $$\omega_n = 2.3 \text{ rad/sec}$$ you can solve the following system of $$2$$ equations:

\begin{align} p_1+p_2 &= 2\zeta \omega_n \\ p_1 \ \cdot \ p_2 &= \omega_{n}^{2} \end{align}

I strongly encourage you to firstly try these by hand and after that you can always use software like MATLAB or Mathematica in order to validate your results.