Nichols plot

The Nichols plot is a plot used in signal processing and control design, named after American engineer Nathaniel B. Nichols. The Nichols plot is a plot used in signal processing and control design, named after American engineer Nathaniel B. Nichols. Given a transfer function, G ( s ) = Y ( s ) X ( s ) {displaystyle G(s)={frac {Y(s)}{X(s)}}} with the closed-loop transfer function defined as, M ( s ) = G ( s ) 1 + G ( s ) {displaystyle M(s)={frac {G(s)}{1+G(s)}}} the Nichols plots displays 20 log 10 ⁡ ( | G ( s ) | ) {displaystyle 20log _{10}(|G(s)|)} versus arg ⁡ ( G ( s ) ) {displaystyle arg(G(s))} . Loci of constant 20 log 10 ⁡ ( | M ( s ) | ) {displaystyle 20log _{10}(|M(s)|)} and arg ⁡ ( M ( s ) ) {displaystyle 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 {displaystyle omega } 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 ) | ) {displaystyle 20log _{10}(|G(s)|)} versus log 10 ⁡ ( ω ) {displaystyle log _{10}(omega )} and arg ⁡ ( G ( s ) ) {displaystyle arg(G(s))} versus log 10 ⁡ ( ω ) {displaystyle log _{10}(omega )} ) - 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 ) ) {displaystyle 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.