स्टेट-स्पेस विश्लेषण में स्थिरता के संबंध में निम्नलिखित कथनों पर विचार कीजिए:
1. सिस्टम मैट्रिक्स A के आइगेनमान अभिलाक्षणिक मूल परिभाषित करते हैं जो सिस्टम की स्थिरता निर्धारित करते हैं।
2. सिस्टम तभी और केवल तभी उपगामी रूप से स्थिर (asymptotically stable) है जब A के सभी आइगेनमानों के वास्तविक भाग पूर्णतः ऋणात्मक हों (बायाँ अर्ध-प्लेन)।
3. धनात्मक वास्तविक भाग वाला कम से कम एक आइगेनमान (दायाँ अर्ध-प्लेन) होने का अर्थ है कि सिस्टम अस्थिर है।
4. स्थिरता सिस्टम का स्वयं का गुण है और आंतरिक पैरामीटर तथा A मैट्रिक्स द्वारा निर्धारित होती है।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
सतत-समय LTI मॉडल की स्थिरता A के आइगेनमानों से जुड़ी है: सभी के वास्तविक भाग पूर्णतः ऋणात्मक हों तो उपगामी स्थिरता; दाएँ अर्ध-प्लेन में एक भी आइगेनमान हो तो अस्थिरता। स्थिरता A द्वारा निर्धारित आंतरिक गुण है। मुख्य संबंध: ẋ=Ax+Bu।
परीक्षा फोकस: A के आइगेनमान = अभिलाक्षणिक मूल।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
Eigenvalues of the system matrix A define the characteristic roots that determine the system's stability.
Eigenvalues of A are the state-model characteristic roots. - Statement 2 - Correct.
A system is asymptotically stable if and only if, all eigenvalues of the system matrix A have strictly negative real parts (located in the Left-Half Plane).
Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. - Statement 3 - Correct.
The presence of at least one eigenvalue with a positive real part (located in the Right-Half Plane) implies that the system is unstable.
Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. - Statement 4 - Correct.
Stability is a property of the system itself and is determined by the internal system parameters and the A matrix.
A state-space model represents internal dynamics using a minimal set of state variables.
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) 4. Option B is incorrect because it omits correct statement(s) 2, 4. Option C is incorrect because it omits correct statement(s) 1.
Exam Shortcut / Approach
Eigenvalues of A are the state-model characteristic roots.
Common Mistake
Output is not generally independent of state variables.
Quick Revision
Eigenvalue Criterion for Stability is linked with Stability, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.
Consider the following statements regarding Stability in State-Space Analysis:
1. Eigenvalues of the system matrix A define the characteristic roots that determine the system's stability.
2. A system is asymptotically stable if and only if, all eigenvalues of the system matrix A have strictly negative real parts (located in the Left-Half Plane).
3. The presence of at least one eigenvalue with a positive real part (located in the Right-Half Plane) implies that the system is unstable.
4. Stability is a property of the system itself and is determined by the internal system parameters and the A matrix.
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. Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. 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.
Eigenvalues of the system matrix A define the characteristic roots that determine the system's stability.
Eigenvalues of A are the state-model characteristic roots. - Statement 2 - Correct.
A system is asymptotically stable if and only if, all eigenvalues of the system matrix A have strictly negative real parts (located in the Left-Half Plane).
Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. - Statement 3 - Correct.
The presence of at least one eigenvalue with a positive real part (located in the Right-Half Plane) implies that the system is unstable.
Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. - Statement 4 - Correct.
Stability is a property of the system itself and is determined by the internal system parameters and the A matrix.
A state-space model represents internal dynamics using a minimal set of state variables.
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) 4. Option B is incorrect because it omits correct statement(s) 2, 4. Option C is incorrect because it omits correct statement(s) 1.
Exam Shortcut / Approach
Eigenvalues of A are the state-model characteristic roots.
Common Mistake
Output is not generally independent of state variables.
Quick Revision
Eigenvalue Criterion for Stability is linked with Stability, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.
સ્ટેટ-સ્પેસ વિશ્લેષણમાં સ્થિરતા સંબંધિત નીચેના વિધાનો ધ્યાનમાં લો:
1. સિસ્ટમ મેટ્રિક્સ A ના આઇગેનમૂલ્યો લાક્ષણિક મૂળ વ્યાખ્યાયિત કરે છે જે સિસ્ટમની સ્થિરતા નક્કી કરે છે.
2. સિસ્ટમ ત્યારે અને ફક્ત ત્યારે જ ઉપગામી રીતે સ્થિર (asymptotically stable) છે જ્યારે A ના બધાં આઇગેનમૂલ્યોના વાસ્તવિક ભાગ સંપૂર્ણ ઋણ હોય (ડાબું અર્ધ-પ્લેન).
3. ધન વાસ્તવિક ભાગ ધરાવતું ઓછામાં ઓછું એક આઇગેનમૂલ્ય (જમણું અર્ધ-પ્લેન) હોય તો સિસ્ટમ અસ્થિર છે.
4. સ્થિરતા સિસ્ટમનો પોતાનો ગુણધર્મ છે અને આંતરિક પરિમાણો તથા A મેટ્રિક્સ દ્વારા નક્કી થાય છે.
ઉપરોક્તમાંથી કયું/કયા વિધાન સાચું/સાચા છે?
જવાબ અને પગલાવાર ઉકેલ
સતત-સમય LTI મોડેલની સ્થિરતા A ના આઇગેનમૂલ્યો સાથે જોડાયેલી છે: બધાંના વાસ્તવિક ભાગ સંપૂર્ણ ઋણ હોય તો ઉપગામી સ્થિરતા; જમણા અર્ધ-પ્લેનમાં એક પણ આઇગેનમૂલ્ય હોય તો અસ્થિરતા. સ્થિરતા A દ્વારા નક્કી થતો આંતરિક ગુણધર્મ છે. મુખ્ય સંબંધ: ẋ=Ax+Bu.
પરીક્ષા મુદ્દો: A ના આઇગેનમૂલ્યો = લાક્ષણિક મૂળ.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
Eigenvalues of the system matrix A define the characteristic roots that determine the system's stability.
Eigenvalues of A are the state-model characteristic roots. - Statement 2 - Correct.
A system is asymptotically stable if and only if, all eigenvalues of the system matrix A have strictly negative real parts (located in the Left-Half Plane).
Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. - Statement 3 - Correct.
The presence of at least one eigenvalue with a positive real part (located in the Right-Half Plane) implies that the system is unstable.
Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. - Statement 4 - Correct.
Stability is a property of the system itself and is determined by the internal system parameters and the A matrix.
A state-space model represents internal dynamics using a minimal set of state variables.
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) 4. Option B is incorrect because it omits correct statement(s) 2, 4. Option C is incorrect because it omits correct statement(s) 1.
Exam Shortcut / Approach
Eigenvalues of A are the state-model characteristic roots.
Common Mistake
Output is not generally independent of state variables.
Quick Revision
Eigenvalue Criterion for Stability is linked with Stability, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.
More State-Space Analysis questions