Equations Outline Discretetime state equation from solution of continuoustime state equation Expressions in terms of constituent matrices Example 2 Solution of State Equation ID: 729197
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Slide1
Discrete-Time State-SpaceEquationsSlide2
Outline
• Discrete-time state equation from
solution of
continuous-time state equation.• Expressions in terms of constituent matrices.• Example.
2Slide3
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 ofthe 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
3Slide4
Piece-wise Constant Input
4Slide5
State & Input Matrices
A
d
= discrete state matrixBd = 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 invertiblestate matrix A, integral of the matrix exponential is
5Slide6
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.
6Slide7
Input Matrix
7Slide8
Discrete-time State-spaceRepresentation
• Discrete state & output equation.
• Discrete-time state equation:
approximately valid for a general input vector u(
t
)
providedthat the sampling period T is sufficiently short.8Slide9
Example 7.15
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Discrete state matrix
10Slide11
discrete-time input matrix
11Slide12
MATLAB
MATLAB command to obtain (
A
d , Bd , C,D)
form (
A
, B, C,D)» pd = c2d(p)Alternatively the matrices are obtained usingthe MATLAB commands» ad = expm(a * 0.05)» bd = a\ (ad-eye(3) )* b12