Nichols plot

Use in Control Design

Given a transfer function,

G ( s ) = Y ( s ) X ( s )

with the closed-loop transfer function defined as,

M ( s ) = G ( s ) 1 + G ( s )

the Nichols plots displays 20 log 10 ⁡ ( | G ( s ) | ) versus arg ⁡ ( G ( s ) ) . Loci of constant 20 log 10 ⁡ ( | M ( s ) | ) and arg ⁡ ( M ( s ) ) are overlaid to allow the designer to obtain the closed loop transfer function directly from the open loop transfer function. Thus, the frequency ω is the parameter along the curve. This plot may be compared to the Bode plot in which the two inter-related graphs - 20 log 10 ⁡ ( | G ( s ) | ) versus log 10 ⁡ ( ω ) and arg ⁡ ( G ( s ) ) versus log 10 ⁡ ( ω ) ) - are plotted.

In feedback control design, the plot is useful for assessing the stability and robustness of a linear system. This application of the Nichols plot is central to the Quantitative feedback theory (QFT) of Horowitz and Sidi, which is a well known method for robust control system design.

In most cases, arg ⁡ ( G ( s ) ) refers to the phase of the system's response. Although similar to a Nyquist plot, a Nichols plot is plotted in a Cartesian coordinate system while a Nyquist plot is plotted in a polar coordinate system.