अवस्था चर विश्लेषण (State Variable Analysis) के संबंध में निम्नलिखित कथनों पर विचार कीजिए:
1. अवस्था चर सिस्टम गतिकी को पूर्णतः परिभाषित करते हैं।
2. स्टेट-स्पेस मॉडल MIMO सिस्टम को दर्शा सकता है।
3. स्टेट मॉडल से ट्रांसफर फंक्शन हमेशा अद्वितीय रूप से प्राप्त किया जा सकता है।
4. स्टेट मॉडल में प्रारंभिक स्थितियाँ शामिल होती हैं।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
स्टेट-स्पेस मॉडल न्यूनतम अवस्था चरों से आंतरिक गतिकी दर्शाता है, स्वाभाविक रूप से MIMO सिस्टम संभालता है और प्रारंभिक स्थितियाँ शामिल करता है। मुख्य संबंध: ẋ=Ax+Bu; y=Cx+Du।
परीक्षा फोकस: A के आइगेनमान अभिलाक्षणिक मूल हैं।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
State variables define system dynamics completely.
A state-space model represents internal dynamics using a minimal set of state variables. - Statement 2 - Correct.
State-space model can represent MIMO systems.
A state-space model represents internal dynamics using a minimal set of state variables. - Statement 3 - Incorrect.
Transfer function can always be uniquely obtained from state model.
A transfer function is an input-output description; different state realizations can represent the same transfer function, and internal-state information is not unique from the transfer function - Statement 4 - Correct.
State model includes initial conditions.
Eigenvalues of A are the state-model characteristic roots.
Core Concept
A state-space model represents internal dynamics using a minimal set of state variables. It naturally handles MIMO systems and incorporates initial conditions. Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. Output generally depends on both state and direct input through y=Cx+Du.
Formula / Key Relationship
ẋ=Ax+Bu; y=Cx+Du.
Why the Other Options Are Wrong
Option A is incorrect because it includes incorrect statement(s) 3. Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 2, 4. Option C is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4.
Exam Shortcut / Approach
Eigenvalues of A are the state-model characteristic roots.
Common Mistake
Output is not generally independent of state variables.
Quick Revision
State Model MIMO and Initial Conditions is linked with State Variables, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.
Consider the following statements regarding State Variable Analysis:
1. State variables define system dynamics completely.
2. State-space model can represent MIMO systems.
3. Transfer function can always be uniquely obtained from state model.
4. State model includes initial conditions.
Which of the above statements is/are correct?
Answer & step-by-step solution
A state-space model represents internal dynamics using a minimal set of state variables. It naturally handles MIMO systems and incorporates initial conditions. Key relation: ẋ=Ax+Bu; y=Cx+Du.
Exam focus: Eigenvalues of A are the state-model characteristic roots. Common trap: Output is not generally independent of state variables.
Statement-wise Verification
- Statement 1 - Correct.
State variables define system dynamics completely.
A state-space model represents internal dynamics using a minimal set of state variables. - Statement 2 - Correct.
State-space model can represent MIMO systems.
A state-space model represents internal dynamics using a minimal set of state variables. - Statement 3 - Incorrect.
Transfer function can always be uniquely obtained from state model.
A transfer function is an input-output description; different state realizations can represent the same transfer function, and internal-state information is not unique from the transfer function - Statement 4 - Correct.
State model includes initial conditions.
Eigenvalues of A are the state-model characteristic roots.
Core Concept
A state-space model represents internal dynamics using a minimal set of state variables. It naturally handles MIMO systems and incorporates initial conditions. Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. Output generally depends on both state and direct input through y=Cx+Du.
Formula / Key Relationship
ẋ=Ax+Bu; y=Cx+Du.
Why the Other Options Are Wrong
Option A is incorrect because it includes incorrect statement(s) 3. Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 2, 4. Option C is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4.
Exam Shortcut / Approach
Eigenvalues of A are the state-model characteristic roots.
Common Mistake
Output is not generally independent of state variables.
Quick Revision
State Model MIMO and Initial Conditions is linked with State Variables, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.
અવસ્થા ચલ વિશ્લેષણ (State Variable Analysis) સંબંધિત નીચેના વિધાનો ધ્યાનમાં લો:
1. અવસ્થા ચલો સિસ્ટમની ગતિશીલતાને સંપૂર્ણ રીતે વ્યાખ્યાયિત કરે છે.
2. સ્ટેટ-સ્પેસ મોડેલ MIMO સિસ્ટમને દર્શાવી શકે છે.
3. સ્ટેટ મોડેલ પરથી ટ્રાન્સફર ફંક્શન હંમેશા અનન્ય રીતે મેળવી શકાય છે.
4. સ્ટેટ મોડેલમાં પ્રારંભિક સ્થિતિઓ સમાવિષ્ટ હોય છે.
ઉપરોક્તમાંથી કયું/કયા વિધાન સાચું/સાચા છે?
જવાબ અને પગલાવાર ઉકેલ
સ્ટેટ-સ્પેસ મોડેલ લઘુતમ અવસ્થા ચલો વડે આંતરિક ગતિશીલતા દર્શાવે છે, સ્વાભાવિક રીતે MIMO સિસ્ટમ સંભાળે છે અને પ્રારંભિક સ્થિતિઓ સમાવે છે. મુખ્ય સંબંધ: ẋ=Ax+Bu; y=Cx+Du.
પરીક્ષા મુદ્દો: A ના આઇગેનમૂલ્યો લાક્ષણિક મૂળ છે.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
State variables define system dynamics completely.
A state-space model represents internal dynamics using a minimal set of state variables. - Statement 2 - Correct.
State-space model can represent MIMO systems.
A state-space model represents internal dynamics using a minimal set of state variables. - Statement 3 - Incorrect.
Transfer function can always be uniquely obtained from state model.
A transfer function is an input-output description; different state realizations can represent the same transfer function, and internal-state information is not unique from the transfer function - Statement 4 - Correct.
State model includes initial conditions.
Eigenvalues of A are the state-model characteristic roots.
Core Concept
A state-space model represents internal dynamics using a minimal set of state variables. It naturally handles MIMO systems and incorporates initial conditions. Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. Output generally depends on both state and direct input through y=Cx+Du.
Formula / Key Relationship
ẋ=Ax+Bu; y=Cx+Du.
Why the Other Options Are Wrong
Option A is incorrect because it includes incorrect statement(s) 3. Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 2, 4. Option C is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4.
Exam Shortcut / Approach
Eigenvalues of A are the state-model characteristic roots.
Common Mistake
Output is not generally independent of state variables.
Quick Revision
State Model MIMO and Initial Conditions is linked with State Variables, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.
More State-Space Analysis questions