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GATE 2014 EC (EC3) – Question 57

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

An unforced system obeys $\dot{\mathbf x}=A\mathbf x$. Initial state $(1,-1)^T$ gives $\mathbf x(t)=(e^{-t},-e^{-t})^T$; initial state $(0,1)^T$ gives $(e^{-t}-e^{-2t},-e^{-t}+2e^{-2t})^T$. Find the response to initial state $(3,5)^T$.

  1. $\begin{pmatrix}-8e^{-t}+11e^{-2t}\\8e^{-t}-22e^{-2t}\end{pmatrix}$
  2. $\begin{pmatrix}11e^{-t}-8e^{-2t}\\-11e^{-t}+16e^{-2t}\end{pmatrix}$
  3. $\begin{pmatrix}3e^{-t}-5e^{-2t}\\-3e^{-t}+10e^{-2t}\end{pmatrix}$
  4. $\begin{pmatrix}5e^{-t}-3e^{-2t}\\-5e^{-t}+6e^{-2t}\end{pmatrix}$

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Correct answer: (B) $\begin{pmatrix}11e^{-t}-8e^{-2t}\\-11e^{-t}+16e^{-2t}\end{pmatrix}$

Explanation

Decompose(3,5)=3(1,−1)+8(0,1). The unforced linear system response is the same combination of the two supplied responses. Thus x1=3e⁻ᵗ+8(e⁻ᵗ−e⁻²ᵗ)=11e⁻ᵗ−8e⁻²ᵗ and x2=−3e⁻ᵗ+8(−e⁻ᵗ+2e⁻²ᵗ)=−11e⁻ᵗ+16e⁻²ᵗ, **B**.