Find K1 and K2 so that peak overshoot = 0.25 and peak time is 2 seconds when unit step input is applied.
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The Correct Answer and Explanation is:
To solve for K1K_1 and K2K_2, we use the standard second-order system equations.
Given that peak time TpT_p is 2 seconds and maximum overshoot MpM_p is 0.25, we use the following relations:
- Peak time equation:
Tp=πωdT_p = \frac{\pi}{\omega_d}
where ωd=ωn1−ζ2\omega_d = \omega_n \sqrt{1 – \zeta^2}.
- Maximum overshoot equation:
Mp=e−πζ1−ζ2M_p = e^{\frac{-\pi \zeta}{\sqrt{1 – \zeta^2}}}
Solving for damping ratio ζ\zeta,
0.25=e−πζ1−ζ20.25 = e^{\frac{-\pi \zeta}{\sqrt{1 – \zeta^2}}}
Taking natural logarithm,
ln(0.25)=−πζ1−ζ2\ln(0.25) = \frac{-\pi \zeta}{\sqrt{1 – \zeta^2}}
Solving for ζ\zeta, we get approximately 0.404.
- Substituting Tp=2T_p = 2 into the peak time equation,
2=πωn1−0.40422 = \frac{\pi}{\omega_n \sqrt{1 – 0.404^2}}
Solving for ωn\omega_n, we get 2.61 rad/s.
For the second problem, we equate given system parameters to the standard second-order system equation:
ωn=5,ζ=0.7\omega_n = 5, \quad \zeta = 0.7
Solving for the system parameters KK and aa, we match them with standard equations, obtaining K=25K = 25 and a=7a = 7.
Explanation
Control systems are analyzed using standard second-order system equations that define transient response characteristics. Peak overshoot indicates how much the system output exceeds the desired steady-state value, while peak time determines how quickly the system reaches this first peak. The damping ratio and natural frequency influence system stability and responsiveness.
Using logarithmic transformations, we solve for the damping ratio from the overshoot equation. Substituting into the peak time equation helps determine the system’s natural frequency. These relationships enable fine-tuning of control parameters to achieve desired performance characteristics.
For the second problem, damping ratio and natural frequency provide constraints. By relating them to standard transfer functions, we derive system parameters that meet specified criteria, ensuring stable and predictable response.
