A system is represented in state-space form with two state variables and stable eigenvalues, then
1. The system order is two.
2. The system is stable since eigenvalues lie in left half-plane.
3. State variables completely describe system dynamics.
4. Output depends only on input and not on state variables.
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. Output generally depends on both state and direct input through y=Cx+Du. 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.
The system order is two.
Ẋ=Ax+Bu; y=Cx+Du. - Statement 2 - Correct.
The system is stable since eigenvalues lie in left half-plane.
Eigenvalues of A are the state-model characteristic roots. - Statement 3 - Correct.
State variables completely describe system dynamics.
A state-space model represents internal dynamics using a minimal set of state variables. - Statement 4 - Incorrect.
Output depends only on input and not on state variables.
State-space output is generally y=Cx+Du and therefore depends on states
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 omits correct statement(s) 1. Option B is incorrect because it omits correct statement(s) 2. Option D is incorrect because it includes incorrect statement(s) 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
System Order Eigenvalues and Output Equation is linked with State Variables, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.
एक सिस्टम को दो अवस्था चरों (state variables) और स्थिर आइगेनमानों के साथ स्टेट-स्पेस रूप में दर्शाया गया है, तब
1. सिस्टम की कोटि दो है।
2. आइगेनमान बाएँ अर्ध-प्लेन में हैं इसलिए सिस्टम स्थिर है।
3. अवस्था चर सिस्टम गतिकी का पूर्ण वर्णन करते हैं।
4. आउटपुट केवल इनपुट पर निर्भर करता है, अवस्था चरों पर नहीं।
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। आउटपुट सामान्यतः अवस्था और इनपुट दोनों पर निर्भर करता है: y=Cx+Du। मुख्य संबंध: ẋ=Ax+Bu; y=Cx+Du।
परीक्षा फोकस: A के आइगेनमान ही अभिलाक्षणिक मूल हैं। सामान्य गलती: आउटपुट सामान्यतः अवस्था चरों से स्वतंत्र नहीं होता।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
The system order is two.
Ẋ=Ax+Bu; y=Cx+Du. - Statement 2 - Correct.
The system is stable since eigenvalues lie in left half-plane.
Eigenvalues of A are the state-model characteristic roots. - Statement 3 - Correct.
State variables completely describe system dynamics.
A state-space model represents internal dynamics using a minimal set of state variables. - Statement 4 - Incorrect.
Output depends only on input and not on state variables.
State-space output is generally y=Cx+Du and therefore depends on states
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 omits correct statement(s) 1. Option B is incorrect because it omits correct statement(s) 2. Option D is incorrect because it includes incorrect statement(s) 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
System Order Eigenvalues and Output Equation is linked with State Variables, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.
એક સિસ્ટમને બે અવસ્થા ચલો (state variables) અને સ્થિર આઇગેનમૂલ્યો સાથે સ્ટેટ-સ્પેસ સ્વરૂપમાં દર્શાવેલ છે, તો
1. સિસ્ટમનો ક્રમ બે છે.
2. આઇગેનમૂલ્યો ડાબા અર્ધ-પ્લેનમાં છે તેથી સિસ્ટમ સ્થિર છે.
3. અવસ્થા ચલો સિસ્ટમની ગતિશીલતાનું સંપૂર્ણ વર્ણન કરે છે.
4. આઉટપુટ ફક્ત ઇનપુટ પર આધાર રાખે છે, અવસ્થા ચલો પર નહીં.
ઉપરોક્તમાંથી કયું/કયા વિધાન સાચું/સાચા છે?
જવાબ અને પગલાવાર ઉકેલ
સ્ટેટ-સ્પેસ મોડેલ લઘુતમ અવસ્થા ચલો વડે આંતરિક ગતિશીલતા દર્શાવે છે. આઉટપુટ સામાન્ય રીતે અવસ્થા અને ઇનપુટ બંને પર આધાર રાખે છે: y=Cx+Du. મુખ્ય સંબંધ: ẋ=Ax+Bu; y=Cx+Du.
પરીક્ષા મુદ્દો: A ના આઇગેનમૂલ્યો જ લાક્ષણિક મૂળ છે. સામાન્ય ભૂલ: આઉટપુટ સામાન્ય રીતે અવસ્થા ચલોથી સ્વતંત્ર નથી.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
The system order is two.
Ẋ=Ax+Bu; y=Cx+Du. - Statement 2 - Correct.
The system is stable since eigenvalues lie in left half-plane.
Eigenvalues of A are the state-model characteristic roots. - Statement 3 - Correct.
State variables completely describe system dynamics.
A state-space model represents internal dynamics using a minimal set of state variables. - Statement 4 - Incorrect.
Output depends only on input and not on state variables.
State-space output is generally y=Cx+Du and therefore depends on states
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 omits correct statement(s) 1. Option B is incorrect because it omits correct statement(s) 2. Option D is incorrect because it includes incorrect statement(s) 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
System Order Eigenvalues and Output Equation is linked with State Variables, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.
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