निम्नलिखित कथन पढ़िए:
कथन 1: यदि नियंत्रण सदिश u(t) द्वारा निर्दिष्ट परिमित समय में सिस्टम अवस्था को प्रारंभिक अवस्था X(t_0) से किसी भी वांछित अवस्था X(t) तक ले जाना संभव हो, तो सिस्टम पूर्णतः नियंत्रणीय (controllable) कहलाता है।
कथन 2: यदि समय अंतराल 0 ≤ t ≤ ∞ में आउटपुट y(t) के मापन से प्रत्येक अवस्था X(t_0) को पूर्णतः पहचाना जा सके, तो सिस्टम पूर्णतः प्रेक्षणीय (observable) कहलाता है।
निम्नलिखित में से सही विकल्प चुनिए:
उत्तर और चरणबद्ध हल
कथन 1 पूर्ण नियंत्रणीयता का मानक अर्थ है: इनपुट परिमित समय में अवस्था को किसी भी वांछित अवस्था तक ले जा सकता है। कथन 2 प्रेक्षणीयता का विचार है: आउटपुट मापन से प्रारंभिक अवस्था पुनः प्राप्त की जा सकती है। n-अवस्था LTI मॉडल के लिए rank(𝒞)=n और rank(𝒪)=n। अतः विकल्प A।
विस्तृत हल (मूल भाषा में)
What This Question Is Really Testing
State variables represent the internal condition of a dynamic system. Controllability asks whether the input can move that internal state where we want it. Observability asks whether output measurements contain enough information to determine the internal state.
Concept Foundation
For the LTI state model ẋ = Ax + Bu, y = Cx + Du, the two standard matrix tests are:
𝒞 = [ B AB A²B … A^n−1B ]
𝒪 = [ C ; CA ; CA² ; … ; CA^n−1 ]
Step 1 — Evaluate Statement 1: Controllability
The statement says that a suitable control input can transfer the system from an initial state to any desired state in a specified finite time. That is the defining idea of complete controllability .
rank(𝒞) = n ⇒ completely controllable
Step 2 — Evaluate Statement 2: Observability
The statement says that the state can be completely identified from measured outputs over the observation interval. That is the defining idea of complete observability .
rank(𝒪) = n ⇒ completely observable
Step 3 — Select the Statement Combination
Both definitions express the intended properties correctly, so the correct option is A .
Why This Method Works
The direction of information/action distinguishes the concepts: controllability is input → state ; observability is state → measured output .
Independent Verification / Cross-Check
If A, B and C matrices were supplied, the rank tests above would provide an independent algebraic verification. Here the item asks only about the definitions.
Concept Extension
- A mode can be uncontrollable even if it appears in the state equations.
- A mode can be unobservable even though it affects internal state dynamics.
- Minimal realizations are both controllable and observable.
Exam Strategy
Use the mnemonic: Can I command the state? → controllability. Can I see/reconstruct the state? → observability.
Quick Trick
Command it = controllability; see it = observability.
Why Other Options Are Wrong
B and C reject one correct definition; D rejects both. Only A accepts the two intended definitions together.
Common Mistake
Interchanging the matrix tests: B generates the controllability matrix, while C generates the observability matrix.
Read the following statements:
Statement 1: A system is said to be completely controllable if it is possible to transfer the system state from an initial state X(t_0) to any desired state X(t) in a specified finite time by a control vector u(t).
Statement 2: A system is said to be completely observable, if every state X(t_0) can be completely identified by measurement of the outputs y(t) over the time interval 0 ≤ t ≤ ∞.
Choose the correct options from the following:
Answer & step-by-step solution
Statement 1 gives the standard meaning of complete controllability: the input can drive the state to any desired state in finite time. Statement 2 gives the observability idea: the initial state can be reconstructed from output measurements. For an n-state LTI model, the algebraic checks are rank(𝒞)=n and rank(𝒪)=n. Hence option A.
What This Question Is Really Testing
State variables represent the internal condition of a dynamic system. Controllability asks whether the input can move that internal state where we want it. Observability asks whether output measurements contain enough information to determine the internal state.
Concept Foundation
For the LTI state model ẋ = Ax + Bu, y = Cx + Du, the two standard matrix tests are:
𝒞 = [ B AB A²B … A^n−1B ]
𝒪 = [ C ; CA ; CA² ; … ; CA^n−1 ]
Step 1 — Evaluate Statement 1: Controllability
The statement says that a suitable control input can transfer the system from an initial state to any desired state in a specified finite time. That is the defining idea of complete controllability .
rank(𝒞) = n ⇒ completely controllable
Step 2 — Evaluate Statement 2: Observability
The statement says that the state can be completely identified from measured outputs over the observation interval. That is the defining idea of complete observability .
rank(𝒪) = n ⇒ completely observable
Step 3 — Select the Statement Combination
Both definitions express the intended properties correctly, so the correct option is A .
Why This Method Works
The direction of information/action distinguishes the concepts: controllability is input → state ; observability is state → measured output .
Independent Verification / Cross-Check
If A, B and C matrices were supplied, the rank tests above would provide an independent algebraic verification. Here the item asks only about the definitions.
Concept Extension
- A mode can be uncontrollable even if it appears in the state equations.
- A mode can be unobservable even though it affects internal state dynamics.
- Minimal realizations are both controllable and observable.
Exam Strategy
Use the mnemonic: Can I command the state? → controllability. Can I see/reconstruct the state? → observability.
Quick Trick
Command it = controllability; see it = observability.
Why Other Options Are Wrong
B and C reject one correct definition; D rejects both. Only A accepts the two intended definitions together.
Common Mistake
Interchanging the matrix tests: B generates the controllability matrix, while C generates the observability matrix.
નીચેના વિધાનો વાંચો:
વિધાન 1: જો નિયંત્રણ સદિશ u(t) દ્વારા નિર્ધારિત મર્યાદિત સમયમાં સિસ્ટમની અવસ્થાને પ્રારંભિક અવસ્થા X(t_0) થી કોઈ પણ ઇચ્છિત અવસ્થા X(t) સુધી લઈ જવી શક્ય હોય, તો સિસ્ટમ સંપૂર્ણ નિયંત્રણક્ષમ (controllable) કહેવાય.
વિધાન 2: જો સમય અંતરાલ 0 ≤ t ≤ ∞ માં આઉટપુટ y(t) ના માપન પરથી દરેક અવસ્થા X(t_0) સંપૂર્ણ રીતે ઓળખી શકાય, તો સિસ્ટમ સંપૂર્ણ અવલોકનક્ષમ (observable) કહેવાય.
નીચેનામાંથી સાચો વિકલ્પ પસંદ કરો:
જવાબ અને પગલાવાર ઉકેલ
વિધાન 1 સંપૂર્ણ નિયંત્રણક્ષમતાનો પ્રમાણભૂત અર્થ છે: ઇનપુટ મર્યાદિત સમયમાં અવસ્થાને કોઈ પણ ઇચ્છિત અવસ્થા સુધી લઈ જઈ શકે. વિધાન 2 અવલોકનક્ષમતાનો વિચાર છે: આઉટપુટ માપન પરથી પ્રારંભિક અવસ્થા ફરી મેળવી શકાય. n-અવસ્થા LTI મોડેલ માટે rank(𝒞)=n અને rank(𝒪)=n. તેથી વિકલ્પ A.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
What This Question Is Really Testing
State variables represent the internal condition of a dynamic system. Controllability asks whether the input can move that internal state where we want it. Observability asks whether output measurements contain enough information to determine the internal state.
Concept Foundation
For the LTI state model ẋ = Ax + Bu, y = Cx + Du, the two standard matrix tests are:
𝒞 = [ B AB A²B … A^n−1B ]
𝒪 = [ C ; CA ; CA² ; … ; CA^n−1 ]
Step 1 — Evaluate Statement 1: Controllability
The statement says that a suitable control input can transfer the system from an initial state to any desired state in a specified finite time. That is the defining idea of complete controllability .
rank(𝒞) = n ⇒ completely controllable
Step 2 — Evaluate Statement 2: Observability
The statement says that the state can be completely identified from measured outputs over the observation interval. That is the defining idea of complete observability .
rank(𝒪) = n ⇒ completely observable
Step 3 — Select the Statement Combination
Both definitions express the intended properties correctly, so the correct option is A .
Why This Method Works
The direction of information/action distinguishes the concepts: controllability is input → state ; observability is state → measured output .
Independent Verification / Cross-Check
If A, B and C matrices were supplied, the rank tests above would provide an independent algebraic verification. Here the item asks only about the definitions.
Concept Extension
- A mode can be uncontrollable even if it appears in the state equations.
- A mode can be unobservable even though it affects internal state dynamics.
- Minimal realizations are both controllable and observable.
Exam Strategy
Use the mnemonic: Can I command the state? → controllability. Can I see/reconstruct the state? → observability.
Quick Trick
Command it = controllability; see it = observability.
Why Other Options Are Wrong
B and C reject one correct definition; D rejects both. Only A accepts the two intended definitions together.
Common Mistake
Interchanging the matrix tests: B generates the controllability matrix, while C generates the observability matrix.
More State-Space Analysis questions