Time Response Specifications (Second-Order Systems)

 

Time Response Specifications (Second-Order Systems)

These specifications describe how a system responds to a unit step input and are key performance measures in control system analysis.

1. Delay Time ($t_d$)

• Time taken for the response to reach 50% of the final value for the first time.

• Indicates how quickly the system starts reacting.

• $t_d\approx \frac{1+0.7\zeta }{{\omega }_n}$

2. Rise Time ($t_r$)

• Time taken for the response to rise from 0% (or 10%) to 100% (or 90%) of the final value.

• For underdamped systems, it is the time to go from 0 to the first peak.

• Smaller rise time → faster system.

• $t_r\approx \frac{\pi -\theta }{{\omega }_d}$, where ${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$

• θ=cos⁡−1(ζ)

3. Peak Time ($t_p$)

• Time taken to reach the first maximum peak of the response.

• For underdamped systems:

$$t_p=\frac{\pi }{{\omega }_d},{\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$$

4. Settling Time ($t_s$)

• Time taken for the response to remain within a specified tolerance band (usually ±2% or ±5%) of the final value.

• The settling time formula for a second-order system (2% criterion) is expressed as:

  $$t_s\approx \frac{4}{\zeta \cdot {\omega }_n}$$

  where:

  • $t_s$ = settling time

  • $\zeta$ = damping ratio

  • ${\omega }_n$ = natural frequency

  🔎 Notes on the formula:

  • The factor 4 corresponds to the 2% tolerance band.

  • For a 5% criterion, the factor becomes 3:

  $$t_s\approx \frac{3}{\zeta \cdot {\omega }_n}$$

  • This approximation works well for underdamped systems ($0<\zeta <1$).

5. Peak Overshoot ($M_p$)

• The maximum amount by which the response exceeds the final value, expressed as a percentage.

• Formula:

$$M_p=e^{\frac{-\zeta \pi }{\sqrt{1-{\zeta }^2}}}\times 100\mathrm{\%}$$

• Higher damping ratio → lower overshoot.

6. Steady-State Error ($e_{ss}$)

• The difference between the final output and the desired input as $t\to \mathrm{\infty }$.

• Depends on the type of system (number of integrators in open-loop transfer function).

  • Type 0 system → finite error for step input.

  • Type 1 system → zero error for step input.

  • Type 2 system → zero error for step and ramp input.

Simple Numerical Example

Given: ${\omega }_n=4\text{rad/s},\zeta =0.5$

• Peak time:

$$t_p=\frac{\pi }{{\omega }_d}=\frac{\pi }{4\sqrt{1-0.25}}=\frac{\pi }{4\cdot \sqrt{0.75}}\approx 0.91\text{s}$$

• Settling time (2% criterion):

$$t_s=\frac{4}{\zeta {\omega }_n}=\frac{4}{0.5\cdot 4}=2\text{s}$$

• Peak overshoot:

$$M_p=e^{\frac{-0.5\pi }{\sqrt{1-0.25}}}\times 100\mathrm{\%}\approx 16.3\mathrm{\%}$$

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