एक प्रणाली को राज्य-अंतरिक्ष रूप में दो राज्य चर और स्थिर eigenvalues ​​के साथ दर्शाया जाता है1.सिस्टम क्रम दो है.2.प्रणाली स्थिर है क्योंकि eigenvalues ​​बाएं आधे तल में स्थित हैं।3.राज्य चर पूरी तरह से सिस्टम की गतिशीलता का वर्णन करते हैं।4.आउटपुट केवल इनपुट पर निर्भर करता है न कि स्टेट वेरिएबल्स पर।उपरोक्त में से कौन सा/से कथन सही है/हैं?

Correct Answer: C - 1, 2 and 3 only

Quick Concept Explanation

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 Equationis linked with State Variables, State-Space Analysis and Control Systems. Eigenvalues of A are the state-model characteristic roots.