Power Systems Dynamics Equations Quiz
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Questions and Answers

In the context of state variable technique, what does the transition matrix represent?

  • The input vector
  • The state vector (correct)
  • The output vector
  • The system matrix
  • What is the representation of the zero-input response in a linear system?

  • y(t) = C(t)x(t) + D(u)
  • y(t) = CΦ(t - t0)x(t0) + ∫CΦ(t - τ)Bu(τ)dτ + Du(t)
  • dx = A(t)x(t) + B(t)u(t)
  • x(t) = Φ(t)x(0) + ∫Φ(t - τ)Bu(τ)dτ (correct)
  • What does the expression 'sX(s) - x(0) = AX(s) + BU(s)' represent in the context of linear systems?

  • Zero-state response (correct)
  • Zero-input response
  • State variable representation
  • Non-homogeneous solution
  • What is the function of the transition matrix in relation to the state variable technique?

    <p>Relates the state and input vectors</p> Signup and view all the answers

    In a multivariable system, what does the state vector represent?

    <p>The internal state of the system</p> Signup and view all the answers

    Study Notes

    State Transition Matrix and Homogeneous Systems

    • A linear time-invariant system is represented by the equations: dx(t)/dt = Ax(t) + Bu(t) and y(t) = Cx(t) + Du(t)
    • The behavior of x(t) and y(t) can be analyzed in two parts: homogeneous solution and non-homogeneous solution
    • A homogeneous system is one whose properties are either the same throughout the system or vary continuously from point to point with no discontinuities
    • The principle of homogeneity states that if a system generates an output y(t) for an input x(t), it must produce an output ay(t) for an input ax(t)

    Homogeneous Solution

    • The homogeneous solution x(t) is the solution to the equation x˙(t) = Ax(t)
    • The homogeneous solution is given by x(t) = e^At x(0)
    • The state transition matrix Φ(t) is defined as Φ(t) = e^At
    • The state transition matrix can be used to find the state x(t) at any time t, given the initial state x(0) and the time elapsed t

    Properties of State Transition Matrix

    • Φ(0) = I (identity matrix)
    • Φ(t1 + t2) = Φ(t1) Φ(t2)
    • Φ(-t) = [Φ(t)]^-1
    • Φ(t) is non-singular and hence invertible

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    Description

    Test your understanding of power systems dynamics equations and their solutions. Determine the behavior of x(t) and y(t), and grasp the principle of homogeneity in system responses.

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