If you specify a settling time in the continuous-time root locus, a vertical line appears on the root locus plot at the pole locations associated with the value provided (using a first-order approximation). In the discrete-time case, the constraint is a curved line.

It is required that $$\mathrm{Re}\left\{pole\right\}<-4.6/{T}_{settling}$$ for continuous systems and $$\mathrm{log}(\text{abs}(pole))/{T}_{discrete}<-4.6/{T}_{settling}$$ for discrete systems. This is an approximation of the settling time based on second-order dominant systems.

Specifying percent overshoot in the continuous-time root locus causes two rays, starting at the root locus origin, to appear. These rays are the locus of poles associated with the percent value (using a second-order approximation). In the discrete-time case, the constraint appears as two curves originating at (1,0) and meeting on the real axis in the left-hand plane.

The percent overshoot *p.o* constraint can
be expressed in terms of the damping ratio, as in this equation:

$$p.o.=100{e}^{-\pi \zeta /\sqrt{1-{\zeta}^{2}}}$$

where $$\zeta $$ is the damping ratio.

Specifying a damping ratio in the continuous-time root locus causes two rays, starting at the root locus origin, to appear. These rays are the locus of poles associated with the damping ratio. In the discrete-time case, the constraint appears as curved lines originating at (1,0) and meeting on the real axis in the left-hand plane.

The damping ratio defines a requirement on $$-\mathrm{Re}\left\{pole\right\}/\text{abs}(pole)$$ for continuous systems and on

$$\begin{array}{l}r=\text{abs}(pSys)\\ t=\text{angle}(pSys)\\ c=-\mathrm{log}(r)/\sqrt{{(\mathrm{log}(r))}^{2}+{t}^{2}}\end{array}$$

for discrete systems.

If you specify a natural frequency, a semicircle centered around the root locus origin appears. The radius equals the natural frequency.

The natural frequency defines a requirement on *abs(pole)* for
continuous systems and on

$$\begin{array}{l}r=\text{abs}(pSys)\\ t=\text{angle}(pSys)\\ c=\sqrt{{(\mathrm{log}(r))}^{2}+{t}^{2}}/T{s}_{model}\end{array}$$

for discrete systems.

Specifies an exclusion region in the complex plane, causing a line to appear between the two specified points with a shaded region below the line. The poles must not lie in the shaded region.

Specify a minimum phase and or a minimum gain margin.

You can specify an upper gain limit, which appears as a straight line on the Bode magnitude curve. You must select frequency limits, the upper gain limit in decibels, and the slope in dB/decade.

Specify the lower gain limit in the same fashion as the upper gain limit.

Specify a minimum phase amount.

While displayed graphically at only one location around a multiple of -180 degrees, this requirement applies to phase margin regardless of actual phase (i.e., it is interpreted for all multiples of -180).

Specify a minimum gain margin.

While displayed graphically at only one location around a multiple of -180 degrees, this requirement applies to gain margin regardless of actual phase (i.e., it is interpreted for all multiples of -180).

Specify a peak closed-loop gain at a given location. The specified value can be positive or negative in dB. The constraint follows the curves of the Nichols plot grid, so it is recommended that you have the grid on when using this feature.

While displayed graphically at only one location around a multiple of -180 degrees, this requirement applies to gain margin regardless of actual phase (i.e., it is interpreted for all multiples of -180).

Specifies an exclusion region for the response on the Nichols plot. The response must not pass through the shaded region.

This only applies to the region (phase and gain) drawn.

You can specify an upper time response bound.

You can specify a lower time response bound.

Was this topic helpful?