Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part49

Signals.and.Systems. - State Equations for DiscreteTime Systems 9.8 State Equations for DiscreteTime Systems As with continuous time systems we

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 9 45 Copyright © Orchard Publications State Equations for Discrete Time Systems 9.8 State Equations for Discrete Time Systems As with continuous time systems, we choose state variables for discrete time systems , either from block diagrams that relate the input output information, or directly from a difference equation. Consider the block diagram of Figure 9.16. Figure 9.16. Block diagram for a continuous time system We learned in Chapter 5 that the state equations representing this continuous time system are (9.123) In a discrete time block diagram, the integrator is replaced by a delay device. The analogy between an integrator and a unit delay device is shown in Figure 9.17. Figure 9.17. Analogy between integration and delay devices Example 9.11 The input output relation for a discrete time system is (9.124) where is any input, and is the output. Write the discrete time state equations for this system. u b A C d x + + + + y Σ Σ t d x · x · Ax bu + = yC x d u + = Delay x · t () continuous time discrete time L xt [] s domain z domain t d xn 1 + xn Z + {} Z z 1 1s L x · t yn 3 + 2y n 2 + 5y n 1 + yn +++ un =
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Chapter 9 Discrete Time Systems and the Z Transform 9 46 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Solution: We choose our state variables as the output and the output advanced by one and by two time steps. Thus, we choose the discrete state variables as (9.125) Then, Thus, the state equations are and in matrix form, (9.126) The general form of the solution is (9.127) The discrete time state equations are written in a more compact form as (9.128) We can use the MATLAB c2d function to convert the continuous time state space equation (9.129) to the discrete time state space equation (9.130) x 1 n [] yn = x 2 n yn 1 + = x 3 n yn 2 + = x 3 n1 + yn 3 + = x 2 + + x 3 n == x 1 + + x 2 n x 1 + x 2 n = x 2 + x 3 n = x 3 + 2x 3 n 5x 2 n x 1 n un x 1 + x 2 + x 3 + 010 001 1 5 2 x 1 n x 2 n x 3 n 0 0 1 + = xn A n x0 A i i0 = bi ui + = xn 1 +
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This note was uploaded on 11/20/2009 for the course EE EE 102 taught by Professor Bar during the Fall '09 term at UCLA.

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Signals.and.Systems. - State Equations for DiscreteTime Systems 9.8 State Equations for DiscreteTime Systems As with continuous time systems we

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