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Discrete-Time State-Space Equations

Discrete-Time State-Space Equations. Outline. • Discrete-time state equation from solution of continuous-time state equation. • Expressions in terms of constituent matrices . • Example. Solution of State Equation. • Analog systems with piecewise constant inputs

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Discrete-Time State-Space Equations

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  1. Discrete-Time State-SpaceEquations

  2. Outline • Discrete-time state equation from solution of continuous-time state equation. • Expressions in terms of constituent matrices. • Example.

  3. Solution of State Equation • Analog systems with piecewise constant inputs over a sampling period: relate state variables at the end of each period by a difference equation. • Obtain difference equation from the solution of the analog state, over a sampling period T. • Solution of state equation for initial time t0 = kT, • and final time tf=(k+1)T = (k+1)T

  4. Piece-wise Constant Input

  5. State & Input Matrices Ad = discrete state matrix Bd= discrete input matrix (same orders as their continuous counterparts). The discrete state matrix = state transition matrix for analog system evaluated at the sampling period T. Properties of the matrix exponential: For invertible state matrix A, integral of the matrix exponential is

  6. Constituent Matrices • Use expansion of the matrix exponential in terms of the constituent matrices. • Eigenvalues of discrete state matrix related to those of the analog system.

  7. Input Matrix

  8. Discrete-time State-spaceRepresentation • Discrete state & output equation. • Discrete-time state equation: approximately valid for a general input vector u(t) provided that the sampling period T is sufficiently short.

  9. Example 7.15

  10. Discrete state matrix

  11. discrete-time input matrix

  12. MATLAB MATLAB command to obtain (Ad, Bd, C,D) form (A, B, C,D) » pd = c2d(p) Alternatively the matrices are obtained using the MATLAB commands » ad = expm(a * 0.05) » bd = a\ (ad-eye(3) )* b

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