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Extras: System Identification

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System parameters such as the damping ratio, the natural frequency, and the DC gain can be found using the step response or bode plot.

Estimating the Order of a System

The order and relative degree of a system can be estimated from either the step response or the bode plot. The relative degree of a system is the difference between the order of the denominator over the order of the numerator of the transfer function and is the lowest order the system can be.

Step Response

If the response of the system to a non-zero step input has a zero slope when t = 0, the system must be second order or higher because the system has a relative degree of two or higher.

If the step response shows oscillations, the system must be a second order or higher underdamped system and have a relative degree of two or higher.

Bode Plot

The phase plot can be a good indicator of order. If the phase drops below -90 degrees, the system must be second order or higher. The relative degree of the system has to be at least as great as the number of multiples of -90 degrees achieved asymptotically at the lowest point on the phase plot of the system.

Identifying a System from the Step Response

DC Gain

The DC gain, $K$, is the ratio of the steady state step response to the magnitude of a step input.

Damping Ratio

For an underdamped second order system, the damping ratio can be calculated from the percent overshoot using the following formula:

(1)$$ \zeta = \frac{-\ln \left( Mp \right)}{\sqrt{\pi^2+\ln^2 \left( Mp \right)}} $$

where $Mp$ is the maximum percent overshoot, which can be approximated from the plot of the step response.

Natural Frequency

The natural frequency of an underdamped second order system can be found from the damped natural frequency which can be measured off the plot of the step response and the damping ratio which was calculated above.

(2)$$ \omega_n = \frac{\omega_d}{\sqrt{1-\zeta^2}} $$

where,

(3)$$ \omega_d = \frac{2\pi}{\Delta t} $$

and $\Delta t$ is the time interval between two consecutive peaks on the plot of the step response.

Identifying a System from the Bode Plot

DC Gain

The DC Gain of a system can be calculated from the magnitude of the bode plot when $s=0$.

(4)$$ K = 10^{M(0)/20} $$

where $M(0)$ is the magnitude of the bode plot when $jw=0$.

Natural Frequency

The damping ratio of a system can be found with the DC Gain and the magnitude of the bode plot when the phase plot is -90 degrees.

(5)$$ \zeta=\frac{K}{2\cdot10^{M(-90^{\circ})/20}} $$

Identifying the System Parameters

If the type of system is known, then specific physical parameters may be found from the dynamic metrics determined above.

The general form of the transfer function of a first order system is

(6)$$ G(s) = \frac{b}{(s+a)}=\frac{K}{\tau s+1} $$

The general form of the transfer function of a second order system is

(7)$$ G(s) = \frac{a}{s^2+bs+c} = \frac{K\omega_n^2}{s^2+2\zeta\omega_n+\omega_n^2} $$


Published with MATLAB® 9.2