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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 5 31 Copyright © Orchard Publications Relationship between State Equations and Laplace Transform In (5.114), is the Laplace transform of the input ; then, division of both sides by yields the transfer function (5.115) Example 5.12 In the circuit of Figure 5.11, all initial conditions are zero. Compute the state transition matrix using the Inverse Laplace transform method. Figure 5.11. Circuit for Example 5.12 Solution: For this circuit, and Substitution of given values and rearranging, yields (5.116) Now, we define the state variables and Then, (5.117) and Also, Us () ut Gs Ys ----------- C s I A 1 bd + == e At + R L + C 3 v S t u 0 t = v C t it 1 H 12 F ii L = Ri L L di L dt ------- v C ++ u 0 t = L 3 i L v C 1 + = x 1 i L = x 2 v C = x · 1 L 3 i L v C 1 + x · 2 dv C -------- =

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Chapter 5 State Variables and State Equations 5 32 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (5.118) and thus, or (5.119) Therefore, from (5.117) and (5.119) we obtain the state equations (5.120) and in matrix form, (5.121) By inspection, (5.122) Now, we will find the state transition matrix from (5.123) where Then, We find the Inverse Laplace of each term by partial fraction expansion. Thus, Now, we can find the state variables representing the inductor current and the capacitor voltage from i L C dv C dt -------- 0.5 C == x 1 i L 0.5 C 0.5x · 2 = x · 2 2x 1 = x · 1 3x 1 x 2 1 + = x · 2 1 = x · 1 x · 2 3 1 20 x 1 x 2 1 0 1 + = A 3 1 = e At L 1 sI A () 1 {} = A s0 0s 3 1 s3 + 1 2 s A 1 adj sI A det sI A --------------------------- 1 s 2 3s 2 ++ ------------------------- s1 2s 3 + s + s2 + -------------------------------- 1 + + 2 + + + + + = e L 1 A 1 e t 2e 2t + e t e + t t e
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 5 33 Copyright © Orchard Publications Relationship between State Equations and Laplace Transform using the procedure of Example 5.11. MATLAB provides two very useful functions to convert state space (state equations), to trans- fer function (s domain), and vice versa. The function ss2tf (state space to transfer function) converts the state space equations

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