Assertion (A) : A PI controller introduces a pole in the system thereby increasing the order and type of system by one. Reason (R) : The increase in the type of the system ensures a decrease in the steady state error of the system. Select your answer using the codes given below.
Both (A) and (R) are true, but (R) is not the correct explanation of (A).
Examine the assertion — true. A proportional-plus-integral controller has the transfer function
\(G_c(s)=K_p+\dfrac{K_i}{s}=\dfrac{K_p s+K_i}{s}=K_p\dfrac{\left(s+K_i/K_p\right)}{s}\)
The denominator carries an explicit factor of s, so the controller adds one pole at the origin (together with a finite zero at \(-K_i/K_p\)). A pole at the origin raises the order of the characteristic equation by one and, by definition, raises the type number by one as well.
Examine the reason — true. Increasing the type does reduce the steady-state error. For a unity-feedback system,
\(e_{ss}=\lim_{s\to 0}\dfrac{sR(s)}{1+G(s)H(s)}\)
and each pole at the origin makes the loop gain infinite at DC for one more class of input:
| Type | Step error | Ramp error | Parabolic error |
|---|---|---|---|
| 0 | 1/(1+Kp) | ∞ | ∞ |
| 1 | 0 | 1/Kv | ∞ |
| 2 | 0 | 0 | 1/Ka |
A type 0 plant with a PI controller becomes type 1, so its step error falls from a finite value to exactly zero — the reason PI control is used at all.
Does the reason explain the assertion? No. The assertion is about what the controller does structurally: it adds a pole, so order and type each rise by one. The reason is about the consequence of that change for accuracy. Improving the steady-state error does not explain why a pole at the origin appears — the explanation for that lies in the 1/s term of the integrator. Effect and cause are the wrong way round, so the code is 2.
The cost of the extra pole. A pole at the origin adds −90° of phase at all frequencies, eroding phase margin and pushing the system towards instability; the zero at \(-K_i/K_p\) is placed to claw some of that phase back. This is why PI control improves accuracy but tends to slow the transient response, and why PID adds a derivative term to restore it.
Hence, both (A) and (R) are true, but (R) is not the correct explanation of (A).
Similar Questions
Which of the following statement are correct for PI controller?
A. PI controller increases the system type by 1, therefore it improves the steady state error by one order.
B. PI controller reduce the rise time
C. PI controller increase the bandwidth.
D. PI controller is a high pass filter.
E. PI controller adds a pole at S=0 to the forward path transfer function, hence type of system increased by 1.
Choose the most appropriate answer from the options given below :
Match List I with List II
LIST I (Transfer function) LIST II (Controller) A. $\frac{K_{1}S + K_{2} + K_{3}S^{2}}{S}$ I. P-controller B. K1 II. PI-controller C. $\frac{K_1S + K_2}{S}$ III. PD-controller D. K1 + K2S IV. PID - controller Choose the correct answer from the options given below:
The value of Kp in Proportional, PI, PID, controllers are given
A. For proportional Kp = T/L
B. For PI, Kp = 0.9 T/L
C. For PI, Kp = 1.7 T/L
D. For PID, Kp = 1.2 T/L
E. For PID Kp = 0.9 T/L
Choose the correct answer from the options given below:
Read the following statements regarding PID controller :
(a) The system complexity of a PID controller is less than that of a PI controller.
(b) A PID controller produces no action for any constant error signal.
(c) A PID controller is used to increase the damping factor of the dominant poles of a PI controlled system.
(d) A PID controller is used to decrease the damping factor of the dominant poles of a PI controlled system.
Which of the above statements are correct ?
Consider the following controllers for their system complexity and arrange them in increasing order of complexity.
(A) Proportional controller
(B) Proportional plus derivative controller
(C) Proportional plus integral plus derivative controller
(D) Proportional plus integralChoose the most appropriate answer from the options given below :
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