## Time- and Frequency-Domain Requirements in Control System Designer App

### Root Locus Diagrams

#### Settling Time

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

#### Percent Overshoot

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.

#### 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 $-\mathrm{Re}\left\{pole\right\}/\text{abs}\left(pole\right)$ for continuous systems and on

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

for discrete systems.

#### Natural Frequency

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

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

for discrete systems.

#### Region Constraint

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.

### Open-Loop and Prefilter Bode Diagrams

#### Gain and Phase Margins

Specify a minimum phase and or a minimum gain margin.

#### Upper Gain Limit

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.

#### Lower Gain Limit

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

### Open-Loop Nichols Plots

#### Phase Margin

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).

#### Gain Margin

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).

#### Closed-Loop Peak Gain

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).

#### Gain-Phase Requirement

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.

### Step/Impulse Response Plots

#### Upper Time Response Bound

You can specify an upper time response bound for step and impulse responses.

#### Lower Time Response Bound

You can specify a lower time response bound for step and impulse responses.

#### Step Response Bound

For a step response plot, you can also specify a step response bound design requirement.

To define a step response bound requirement, specify the following step response parameters:

• Final value — Final steady-state value

• Rise time — Time required to reach the specified percentage, % Rise, of the Final value.

• Settling time — Time at which the response enters and stays within the settling percentage, % Settling, of the Final value.

• % Overshoot — Maximum percentage overshoot above the Final value.

• % Undershoot — Maximum percentage undershoot below the Initial value.

In the Control System Designer app, step response plots always use an Initial value and a Step time of `0`.