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If the unit step response of a system is given by c(t) = -1.5 + et + 0.5e-2f, then the system will be
- a)stable
- b)unstable
- c)relatively stable
- d)marginally stable
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If the unit step response of a system is given by c(t) = -1.5 + et+ 0....
Given, step response:
∴ Transfer function is
∴ Transfer function is
Since one of the closed loop pole (s = 1) lies in RH s-plane, therefore given system is unstable.
Most Upvoted Answer
If the unit step response of a system is given by c(t) = -1.5 + et+ 0....
Understanding the Unit Step Response
The given unit step response of the system is:
c(t) = -1.5 + e^t + 0.5e^(-2t)
Analyzing this expression provides insight into the stability of the system.
Components of the Response
- Constant Term (-1.5): This shifts the response but does not affect stability.
- Exponential Term (e^t): This term grows without bound as time (t) increases. It indicates that the system output will keep increasing indefinitely.
- Damped Exponential Term (0.5e^(-2t)): This term decays to zero as time progresses, contributing to stability. However, its presence is overshadowed by the exponential growth from the e^t term.
Stability Analysis
- Stable System: A system is stable if all components of its response either converge to a finite value or oscillate around it without growing unbounded.
- Unstable System: A system is considered unstable if any part of its response diverges to infinity.
In this case, the presence of the e^t term dominates the response, causing it to grow indefinitely. Thus, the system does not converge or oscillate around a finite value.
Conclusion
Since the system's response includes an exponential term that grows unbounded, it is classified as:
- Unstable (Option 'B')
This conclusion aligns with the definition of stability in control systems, emphasizing the importance of analyzing each term in the response to determine the overall behavior of the system.
The given unit step response of the system is:
c(t) = -1.5 + e^t + 0.5e^(-2t)
Analyzing this expression provides insight into the stability of the system.
Components of the Response
- Constant Term (-1.5): This shifts the response but does not affect stability.
- Exponential Term (e^t): This term grows without bound as time (t) increases. It indicates that the system output will keep increasing indefinitely.
- Damped Exponential Term (0.5e^(-2t)): This term decays to zero as time progresses, contributing to stability. However, its presence is overshadowed by the exponential growth from the e^t term.
Stability Analysis
- Stable System: A system is stable if all components of its response either converge to a finite value or oscillate around it without growing unbounded.
- Unstable System: A system is considered unstable if any part of its response diverges to infinity.
In this case, the presence of the e^t term dominates the response, causing it to grow indefinitely. Thus, the system does not converge or oscillate around a finite value.
Conclusion
Since the system's response includes an exponential term that grows unbounded, it is classified as:
- Unstable (Option 'B')
This conclusion aligns with the definition of stability in control systems, emphasizing the importance of analyzing each term in the response to determine the overall behavior of the system.
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If the unit step response of a system is given by c(t) = -1.5 + et+ 0.5e-2f, then the system will bea)stableb)unstablec)relatively stabled)marginally stableCorrect answer is option 'B'. Can you explain this answer?
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