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GATE 2023 EC – Question 43

Control Systems · State Variable Model and Solution of State Equation · 2 marks · Multiple choice

The state equation of a second order system is $\dot{\boldsymbol x}(t)=A\boldsymbol x(t)$, with $\boldsymbol x(0)$ the initial condition. Suppose $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of $A$ and $\boldsymbol v_1$ and $\boldsymbol v_2$ are the corresponding eigenvectors. For constants $\alpha_1$ and $\alpha_2$, the solution, $\boldsymbol x(t)$, of the state equation is

  1. $\sum_{i=1}^2\alpha_ie^{\lambda_it}\boldsymbol v_i$
  2. $\sum_{i=1}^2\alpha_ie^{2\lambda_it}\boldsymbol v_i$
  3. $\sum_{i=1}^2\alpha_ie^{3\lambda_it}\boldsymbol v_i$
  4. $\sum_{i=1}^2\alpha_ie^{4\lambda_it}\boldsymbol v_i$

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Correct answer: (A) $\sum_{i=1}^2\alpha_ie^{\lambda_it}\boldsymbol v_i$

Explanation

Expand $\boldsymbol x(0)=\sum\alpha_i\boldsymbol v_i$ in eigenvectors. Since $e^{At}\boldsymbol v_i=e^{\lambda_it}\boldsymbol v_i$, $\boldsymbol x(t)=e^{At}\boldsymbol x(0)=\sum_i\alpha_ie^{\lambda_it}\boldsymbol v_i$.