એક યુનિટી ફીડબેક સિસ્ટમનું ઓપન-લૂપ ટ્રાન્સફર ફંક્શન એવું છે કે તે ટાઇપ-1 સિસ્ટમ છે અને તેના પર યુનિટ સ્ટેપ ઇનપુટ લગાડેલ છે, તો
1. સ્ટેપ ઇનપુટ માટે સ્થાયી-સ્થિતિ ત્રુટિ શૂન્ય છે.
2. રેમ્પ ઇનપુટ માટે સ્થાયી-સ્થિતિ ત્રુટિ મર્યાદિત છે.
3. સિસ્ટમ ગેઇન વધારવાથી રેમ્પ ઇનપુટની સ્થાયી-સ્થિતિ ત્રુટિ ઘટે છે.
4. ગેઇન વધતાં સ્ટેપ ઇનપુટની સ્થાયી-સ્થિતિ ત્રુટિ વધે છે.
ઉપરોક્તમાંથી કયું/કયા વિધાન સાચું/સાચા છે?
જવાબ અને પગલાવાર ઉકેલ
ટાઇપ-1 માટે Kp=∞ → સ્ટેપ ત્રુટિ=1/(1+Kp)=0; Kv મર્યાદિત → રેમ્પ ત્રુટિ=1/Kv, જે ગેઇન વધારતાં ઘટે છે (ક્લોઝ્ડ લૂપ સ્થિર હોવો જોઈએ). તેથી સાચાં વિધાનો {1,2,3}.
પરીક્ષા મુદ્દો: ટાઇપ-1: સ્ટેપ → શૂન્ય, રેમ્પ → મર્યાદિત, પેરાબોલિક → અનંત ત્રુટિ. સામાન્ય ભૂલ: સ્થિરતાની શરત અવગણવી.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
The steady-state error for step input is zero.
One integrator makes the position constant Kp infinite, so unit-step steady-state error is zero (for a stable closed loop). - Statement 2 - Correct.
The steady-state error for ramp input is finite.
The velocity constant Kv is finite for a type-1 system, so unit-ramp error is finite. - Statement 3 - Correct.
Increasing system gain reduces steady-state error for ramp input.
Increasing system gain typically increases Kv and therefore reduces ess,ramp=1/Kv. - Statement 4 - Incorrect.
The steady-state error for step input increases with increase in gain.
Step error remains zero for a stable type-1 system; it does not increase with gain.
Core Concept
System type is the number of pure integrators in the open-loop transfer function GH(s). It determines the order of polynomial input that can be tracked with zero/finite/infinite steady-state error, provided the closed loop is stable.
Formula / Key Relationship
Type 1: Kp=lim_{s→0}GH=∞ → ess,step=1/(1+Kp)=0; Kv=lim_{s→0}sGH finite → ess,ramp=1/Kv.
Step-by-Step Check
Static-error constants directly produce the truth set {1,2,3}. Option A.
Why the Other Options Are Wrong
Option B is incorrect because it omits correct statement(s) 2. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Answer-Key Verification Note
The uploaded provisional key marks C . After independent technical verification, the defensible answer is A . This discrepancy is stated explicitly rather than silently copying the provisional key.
Exam Shortcut / Approach
Type-1 mnemonic: step → zero error; ramp → finite error; parabolic → infinite error.
Common Mistake
Ignoring the closed-loop stability condition when applying steady-state error formulas.
Quick Revision
Type-1 Error Constants is linked with System Type, Steady-State Analysis and Control Systems. Type-1 mnemonic: step → zero error; ramp → finite error; parabolic → infinite error.
A unity feedback system has open-loop transfer function such that it is a type-1 system and subjected to a unit step input, then
1. The steady-state error for step input is zero.
2. The steady-state error for ramp input is finite.
3. Increasing system gain reduces steady-state error for ramp input.
4. The steady-state error for step input increases with increase in gain.
Which of the above statements is/are correct?
Answer & step-by-step solution
It determines the order of polynomial input that can be tracked with zero/finite/infinite steady-state error, provided the closed loop is stable. Static-error constants directly produce the truth set {1,2,3}. Key relation: Type 1: Kp=lim_{s→0}GH=∞ → ess,step=1/(1+Kp)=0; Kv=lim_{s→0}sGH finite → ess,ramp=1/Kv.
Exam focus: Type-1 mnemonic: step → zero error; ramp → finite error; parabolic → infinite error. Common trap: Ignoring the closed-loop stability condition when applying steady-state error formulas.
Statement-wise Verification
- Statement 1 - Correct.
The steady-state error for step input is zero.
One integrator makes the position constant Kp infinite, so unit-step steady-state error is zero (for a stable closed loop). - Statement 2 - Correct.
The steady-state error for ramp input is finite.
The velocity constant Kv is finite for a type-1 system, so unit-ramp error is finite. - Statement 3 - Correct.
Increasing system gain reduces steady-state error for ramp input.
Increasing system gain typically increases Kv and therefore reduces ess,ramp=1/Kv. - Statement 4 - Incorrect.
The steady-state error for step input increases with increase in gain.
Step error remains zero for a stable type-1 system; it does not increase with gain.
Core Concept
System type is the number of pure integrators in the open-loop transfer function GH(s). It determines the order of polynomial input that can be tracked with zero/finite/infinite steady-state error, provided the closed loop is stable.
Formula / Key Relationship
Type 1: Kp=lim_{s→0}GH=∞ → ess,step=1/(1+Kp)=0; Kv=lim_{s→0}sGH finite → ess,ramp=1/Kv.
Step-by-Step Check
Static-error constants directly produce the truth set {1,2,3}. Option A.
Why the Other Options Are Wrong
Option B is incorrect because it omits correct statement(s) 2. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Answer-Key Verification Note
The uploaded provisional key marks C . After independent technical verification, the defensible answer is A . This discrepancy is stated explicitly rather than silently copying the provisional key.
Exam Shortcut / Approach
Type-1 mnemonic: step → zero error; ramp → finite error; parabolic → infinite error.
Common Mistake
Ignoring the closed-loop stability condition when applying steady-state error formulas.
Quick Revision
Type-1 Error Constants is linked with System Type, Steady-State Analysis and Control Systems. Type-1 mnemonic: step → zero error; ramp → finite error; parabolic → infinite error.
एक यूनिटी फीडबैक सिस्टम का ओपन-लूप ट्रांसफर फंक्शन ऐसा है कि यह टाइप-1 सिस्टम है और इस पर यूनिट स्टेप इनपुट लगाया गया है, तब
1. स्टेप इनपुट के लिए स्थायी-अवस्था त्रुटि शून्य है।
2. रैंप इनपुट के लिए स्थायी-अवस्था त्रुटि परिमित है।
3. सिस्टम गेन बढ़ाने से रैंप इनपुट की स्थायी-अवस्था त्रुटि घटती है।
4. गेन बढ़ने पर स्टेप इनपुट की स्थायी-अवस्था त्रुटि बढ़ती है।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
टाइप-1 के लिए Kp=∞ → स्टेप त्रुटि=1/(1+Kp)=0; Kv परिमित → रैंप त्रुटि=1/Kv, जो गेन बढ़ाने पर घटती है (क्लोज़्ड लूप स्थिर होना चाहिए)। अतः सही कथन {1,2,3}।
परीक्षा फोकस: टाइप-1: स्टेप → शून्य, रैंप → परिमित, पैराबोलिक → अनंत त्रुटि। सामान्य गलती: स्थिरता शर्त को अनदेखा करना।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
The steady-state error for step input is zero.
One integrator makes the position constant Kp infinite, so unit-step steady-state error is zero (for a stable closed loop). - Statement 2 - Correct.
The steady-state error for ramp input is finite.
The velocity constant Kv is finite for a type-1 system, so unit-ramp error is finite. - Statement 3 - Correct.
Increasing system gain reduces steady-state error for ramp input.
Increasing system gain typically increases Kv and therefore reduces ess,ramp=1/Kv. - Statement 4 - Incorrect.
The steady-state error for step input increases with increase in gain.
Step error remains zero for a stable type-1 system; it does not increase with gain.
Core Concept
System type is the number of pure integrators in the open-loop transfer function GH(s). It determines the order of polynomial input that can be tracked with zero/finite/infinite steady-state error, provided the closed loop is stable.
Formula / Key Relationship
Type 1: Kp=lim_{s→0}GH=∞ → ess,step=1/(1+Kp)=0; Kv=lim_{s→0}sGH finite → ess,ramp=1/Kv.
Step-by-Step Check
Static-error constants directly produce the truth set {1,2,3}. Option A.
Why the Other Options Are Wrong
Option B is incorrect because it omits correct statement(s) 2. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Answer-Key Verification Note
The uploaded provisional key marks C . After independent technical verification, the defensible answer is A . This discrepancy is stated explicitly rather than silently copying the provisional key.
Exam Shortcut / Approach
Type-1 mnemonic: step → zero error; ramp → finite error; parabolic → infinite error.
Common Mistake
Ignoring the closed-loop stability condition when applying steady-state error formulas.
Quick Revision
Type-1 Error Constants is linked with System Type, Steady-State Analysis and Control Systems. Type-1 mnemonic: step → zero error; ramp → finite error; parabolic → infinite error.
More Steady-State Error questions