DiscreteTime StateSpace Equations Outline Discretetime state equation from

Discrete-Time State-Space Equations

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

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 t 0 = k. T, • and final time tf =(k+1)T = (k+1)T 3

Piece-wise Constant Input 4

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 5

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. 6

Input Matrix 7

Discrete-time State-space Representation • 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. 8

Example 7. 15 9

Discrete state matrix 10

discrete-time input matrix 11

MATLAB command to obtain (Ad , Bd , C, D) form (A, B, C, D) » pd = c 2 d(p) Alternatively the matrices are obtained using the MATLAB commands » ad = expm(a * 0. 05) » bd = a (ad-eye(3) )* b 12