Read the following statements in the context of a PI controller used to control a first order LTI system.
Statement 1: The order of overall closed loop system becomes higher than the open loop system.
Statement 2: Increasing integral gain makes the overall system to have higher damping.
Choose the correct options from the following:
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Answer & Explanation
Correct AnswerB. Statement 1 is correct but statement 2 is incorrect.
Quick Explanation
A PI controller introduces an integrator 1/s, so a first-order plant normally produces a second-order closed-loop characteristic equation: Statement 1 is true. For a standard plant K/(τs+1) with fixed Kp, ζ = (1 + KKp)/(2√(τKKi)); increasing Ki reduces ζ, so Statement 2 is false.
B — Statement 1 is correct but Statement 2 is incorrect.
The PI integrator increases the characteristic order for a first-order plant, while increasing integral gain alone does not increase damping; under the standard model it reduces the damping ratio.
What This Question Is Really Testing
This question is best answered by deriving a simple closed-loop characteristic equation rather than relying on memorized tuning slogans. A PI controller has both proportional action and an integral pole at the origin.
Concept Foundation
Use a generic first-order plant and PI controller:
G(s) = K/(τs + 1)
C(s) = Kp + Ki/s
C(s) = (Kps + Ki)/s
Step 1 — Form the Closed-Loop Characteristic Equation
For unity negative feedback, the characteristic equation is 1 + C(s)G(s) = 0.
1 + K(Kps + Ki)/[s(τs + 1)] = 0
s(τs + 1) + K(Kps + Ki) = 0
τs² + (1 + KKp)s + KKi = 0
Step 2 — Judge Statement 1
The characteristic polynomial is second order even though the plant itself is first order. Thus the PI integrator has increased the closed-loop dynamic order, so Statement 1 is true (assuming no exact pole-zero cancellation).
Step 3 — Express the Damping Ratio
Divide the characteristic equation by τ and compare it with the standard second-order form.
s² + [(1 + KKp)/τ]s + KKi/τ = 0
s² + 2ζωns + ωn² = 0
ωn = √(KKi/τ)
ζ = (1 + KKp)/(2√(τKKi))
Step 4 — Judge Statement 2
With K, τ and Kp fixed, increasing Ki increases the denominator of the ζ expression, so ζ decreases. Therefore Statement 2 is false.
Step 5 — Select the Option
Statement 1 is true and Statement 2 is false, so the correct option is B.
Why This Method Works
The integrator adds a pole at the origin, which raises the dynamic order. Integral gain strengthens low-frequency error correction, but stronger integral action can make the transient response less damped unless Kp is retuned.
Independent Verification / Cross-Check
The derived formula gives an immediate trend check:
Ki ↑ ⇒ √Ki ↑ ⇒ ζ ↓ (K, τ, Kp fixed)
Concept Extension
PI control is commonly used to eliminate step steady-state error.
Aggressive integral action may increase overshoot and oscillation.
Controller gains should be considered together, not independently.
Exam Strategy
For controller-effect statement questions, derive the characteristic equation of a simple standard plant. It is safer than memorizing “Ki improves X” without conditions.
Quick Trick
PI + first-order plant → normally second-order; larger Kᵢ alone does not mean larger damping.
Why Other Options Are Wrong
A incorrectly accepts the damping statement. C incorrectly rejects the order increase. D rejects the valid first statement.
Common Mistake
Assuming that increasing integral gain automatically improves damping because it improves steady-state accuracy.
Exam Tip
PI adds an integrator. For a first-order plant, expect a second-order characteristic equation unless cancellation is explicitly present.
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