Question
Which of the following statements are CORRECT for a controller? P. In a proportional controller, a control action is proportional to the error Q. In an integral controller, a control action is proportional to the derivative of the error R. There is no “offset” in the response of the closed-loop first-order process with a proportional controller S. There is no "offset" in the response of the closed-loop first-order process with a proportional-integral controller
The correct answer is
P and S only
Controller Statements Analysis
This solution analyzes the correctness of four statements (P, Q, R, S) regarding proportional (P) and integral (I) controllers in control systems.
Statement P: Proportional Controller Action
- A proportional controller produces a control output that is directly proportional to the current error signal.
- Mathematically, the control action $u(t)$ is represented as $u(t) = K_p \cdot e(t)$, where $K_p$ is the proportional gain and $e(t)$ is the error.
- Therefore, statement P is correct.
Statement Q: Integral Controller Action
- An integral controller produces a control output that is proportional to the integral of the error signal over time.
- Mathematically, $u(t) = K_i \int e(t) dt$. The statement describes the action as proportional to the derivative of the error, which corresponds to a derivative controller ($K_d \frac{de(t)}{dt}$).
- Therefore, statement Q is incorrect.
Statement R: Offset with Proportional Controller
- For a standard first-order process controlled by a proportional controller, a steady-state error, known as "offset," typically exists if the setpoint is changed or a load disturbance occurs.
- The offset can only be eliminated if the proportional gain ($K_p$) is infinitely large or if the process itself has an inherent integrating element, which is not generally assumed.
- Therefore, statement R is incorrect.
Statement S: Offset with Proportional-Integral Controller
- A proportional-integral (PI) controller includes both proportional and integral control actions.
- The integral action ($K_i \int e(t) dt$) accumulates past errors. This integral term actively works to drive the steady-state error to zero, thereby eliminating offset in the closed-loop response of typical systems, including first-order processes.
- Therefore, statement S is correct.
Conclusion
Based on the analysis:
- Statement P is correct.
- Statement Q is incorrect.
- Statement R is incorrect.
- Statement S is correct.
The correct statements are P and S. This corresponds to Option 3.
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Important Questions from Proportional Derivative and Integral Control
- A proportional controller is used to control the temperature of an autoclave from $60\degree C$ to $130\degree C$. If the proportional band setting of the controller is 25%, the proportional gain value is ________
- Which one of the following statements is CORRECT about proportional controllers?
- The transfer function of a Proportional-Integral (PI) controller $G_c(s)$ is given by $$G_c(s) = K_c \left(1 + \frac{1}{\tau_I s}\right)$$ where $K_c$ is the controller gain, $\tau_I$ is the controller integral time constant and $s$ is the Laplace variable. The role of the integral component of the controller is to ____________.
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