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Dynamic Systems in State-Space Representation
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Dynamic Systems in State-Space Representation

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@FancyTantalum

Questions and Answers

What is the state-space representation of dynamic systems without feedthrough?

  • An integration of input and output in a single equation
  • A representation that requires multiple equations to describe the system
  • A direct mapping from transfer function to controllable canonical form (correct)
  • A scheme involving a transformation to observable canonical form

    • In the context of state-space control, what does 'feedthrough' refer to?

  • The separation of state variables from system parameters
  • Direct transmission of input to output in a transfer function (correct)
  • An aspect of feedback control systems
  • The process of increasing system stability

    • How does the controllable canonical form change for systems with feedthrough?

  • It simplifies to a first-order system only
  • It requires a reduction of state variables
  • It introduces additional terms into the equations (correct)
  • It eliminates constant terms from the equations

    • Which equation holds true for systems with feedthrough?

    <p>m = n maintains the relationship between input and output</p> Signup and view all the answers

    What is a benefit of using state-space representation?

    <p>It provides a one-to-one relationship with transfer functions</p> Signup and view all the answers

    What happens when the first term is eliminated in the frequency domain for controllable canonical form?

    <p>The form simplifies, incorporating feedthrough dynamics</p> Signup and view all the answers

    What is the engineering view of a matrix in the context of mapping vectors?

    <p>A matrix represents a (linear) mapping of an input vector to an output vector.</p> Signup and view all the answers

    What does it indicate if a matrix has more rows than columns?

    <p>The matrix is considered skinny.</p> Signup and view all the answers

    In which case is a matrix described as 'fat'?

    <p>When it has more columns than rows.</p> Signup and view all the answers

    How are tensors related to matrices according to the computer science view?

    <p>Tensors are simply larger matrices with more than two dimensions.</p> Signup and view all the answers

    What does the term 'dim(x) = n' signify in relation to a matrix?

    <p>It describes the dimensions of the input vector.</p> Signup and view all the answers

    Which of the following correctly describes a square matrix?

    <p>The number of columns equals the number of rows.</p> Signup and view all the answers

    What is one application of a matrix in computer science as mentioned?

    <p>Representing pixels in a digital image.</p> Signup and view all the answers

    If a matrix is described as rectangular, what does that imply about its dimensions?

    <p>The number of columns does not equal the number of rows.</p> Signup and view all the answers

    What is the primary characteristic of a normal case in dynamic systems with regards to feedthrough?

    <p>There is no direct feedthrough from input to output.</p> Signup and view all the answers

    In the context of state-space representation, which symbol represents the scalar feedthrough?

    <p>d</p> Signup and view all the answers

    What condition is met for a system to exhibit direct feedthrough?

    <p>m = n</p> Signup and view all the answers

    Which of the following best describes the output equation in state-space representation?

    <p>It relates the input vector to the output vector through the system matrix.</p> Signup and view all the answers

    What does setting an = 1 imply in the context of the transfer function?

    <p>The function can be normalized.</p> Signup and view all the answers

    Which of the following statements about a dynamic system is true when m < n?

    <p>The system matrix is non-square.</p> Signup and view all the answers

    What is the significance of the n x n system matrix in state-space representation?

    <p>It describes the relationship between state variables.</p> Signup and view all the answers

    Which scenario is most likely to occur infrequently in dynamic systems?

    <p>A system with direct feedthrough.</p> Signup and view all the answers

    What command generates the state-space equations from the matrices A, b, c, and d?

    <p>System = ss(A,b,c,d);</p> Signup and view all the answers

    Which command is used to convert a transfer function's numerator and denominator to state-space form?

    <p>[A,b,c,d] = tf2ss(num,den);</p> Signup and view all the answers

    What is the purpose of the command 'minreal' in the context of dynamic systems?

    <p>To calculate one minimum realization of the system.</p> Signup and view all the answers

    How is the state vector x(t) influenced according to the content?

    <p>It is affected by the initial value x(0) and the input u(t).</p> Signup and view all the answers

    What must be calculated first to subsequently determine the output y(t)?

    <p>The state vector x(t).</p> Signup and view all the answers

    What does the command 'ss2tf' accomplish?

    <p>It converts state-space representation to a transfer function.</p> Signup and view all the answers

    What are the parameters used in the command 'ss(A, b, c, d)' in MATLAB?

    <p>State matrix, input matrix, output matrix, feedforward matrix.</p> Signup and view all the answers

    What does calculating the state vector x(t) allow for in the context of dynamic systems?

    <p>Calculating the output y(t).</p> Signup and view all the answers

    What does the system matrix A represent in the context of state-space control?

    <p>The coefficients of the characteristic equation.</p> Signup and view all the answers

    What is the effect of state feedback on the system matrix A?

    <p>It modifies the poles of the system.</p> Signup and view all the answers

    What is needed to specify the desired characteristic equation in state-space control?

    <p>The n poles 𝜆1, 𝜆2,..., 𝜆n.</p> Signup and view all the answers

    Which statement accurately describes the dyadic product b kT?

    <p>It influences the characteristic equation's coefficients.</p> Signup and view all the answers

    In the context of closed-loop control, what does the characteristic equation relate to?

    <p>The poles defined by the system matrix A.</p> Signup and view all the answers

    What happens to the number of degrees of freedom when implementing state feedback?

    <p>It is equal to n, which is the number of state variables.</p> Signup and view all the answers

    What is the determinant sI – (A – b kT) used for in state-space control?

    <p>To establish the characteristic equation of the closed loop.</p> Signup and view all the answers

    Which of the following is NOT altered by the state feedback in a state-space control system?

    <p>The zeros of the system.</p> Signup and view all the answers

    Study Notes

    Dynamic Systems in State-Space Representation

    • A matrix can represent various concepts; in computer science, it serves as a 2D array, often used to represent pixel data in images, where each element indicates grayscale levels.
    • In engineering, a matrix A maps an input vector x to an output vector y, detailing linear transformations.
    • Matrix dimensions are categorized as rectangular (n ≠ m), fat (n > m), skinny (n < m), or square (n = m).

    State-space Representation of Dynamic Systems

    • An n-th order linear dynamic system can be described via a transfer function, considering two cases based on direct feedthrough: normal (m < n, no direct feedthrough) or rarer cases (m = n, direct feedthrough).
    • State-space equations consist of a system matrix A, input vector b, output vector cT, and potentially a feedthrough scalar d (usually zero when m < n).

    Transfer Function to State-space Conversion

    • There is a direct method to create state equations from transfer functions in controllable canonical form, establishing a 1:1 relationship.
    • For systems with feedthrough, the equations accommodate additional terms that correspond to a different state-space representation.

    MATLAB for State-Space Systems

    • The ss function in MATLAB generates state-space representations by invoking matrices A, b, c, and d.
    • Conversion functions such as tf2ss and ss2tf allow transitions between transfer function forms and state-space gear.
    • The minreal function computes a minimum realization of a dynamic system, optimizing representation.

    Calculating State Vector Evolution

    • Determining how the state vector x(t) evolves over time involves integration with respect to the initial state x(0) and input u(t) to establish the system output y(t) via the output equation.

    State-Space Control via Pole Placement

    • Controller parameters are derived by adjusting the system matrix A through state feedback, replacing the manipulated variable with a reference variable w.
    • Specific pole placement allows control over system dynamics, with feedback mechanisms enabling adjustment of poles in the characteristic equation of the closed loop.
    • The desired characteristic equation for the closed loop is expressed through the determinant sI - (A - bkT), indicating that n poles can be set with k degrees of freedom.

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    Description

    This quiz covers the basics of dynamic systems using state-space representation, focusing on the interpretation of matrices and vectors. You'll explore their significance in computer science and their applications in representing data structures. Test your understanding of these foundational concepts in mathematics.

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