Explanation Given data: The differential equation is, d 3 y ( t ) d t 3 + 18 d 2 y ( t ) d t 2 + 20 d y ( t ) d t + 5 y ( t ) = z ( t ) (1) Formula used: Write a standard expression to represent the state equation. x ˙ = A x + B z (2) Here, A is the system matrix, B is the input coupling matrix, x is the state vector , and z is the input vector. Write a general form to calculate the state vector representing n state vectors. x = [ x 1 x 2 ⋮ x n ] (3) Write a general form to calculate the input vector representing m inputs. z = [ z 1 z 2 ⋮ z m ] (4) Write a general form of output equation. y = C x + D z (5) Here, C is the output matrix, and D is the feed forward matrix. Substitute the equations (3) and (4) in equation (2) to find x ˙ . x ˙ = A [ x 1 x 2 ⋮ x n ] + B [ z 1 z 2 ⋮ z m ] (6) Substitute the equations (3) and (4) in equation (5) to find y . y = C [ x 1 x 2 ⋮ x n ] + D [ z 1 z 2 ⋮ z m ] (7) Calculation: Consider the function x 1 as follows. x 1 = y ( t ) (8) Take the first derivative for equation (8) to find x ˙ 1 . x ˙ 1 = y ˙ ( t ) (9) Consider the first derivative of x 1 is x 2 . x ˙ 1 = x 2 (10) Substitute x 2 for x ˙ 1 in equation (9) to find x 2 . x 2 = y ˙ ( t ) (11) Take the first derivative for equation (11) to find x ˙ 2 . x ˙ 2 = y ¨ ( t ) (12) Consider the first derivative of x 2 is x 3 . x ˙ 2 = x 3 (13) Substitute x 3 for x ˙ 2 in equation (12) to find x 3 . x 3 = y ¨ ( t ) (14) Take the first derivative for equation (14) to find x ˙ 3 . x ˙ 3 = y ⃛ ( t ) (15) Refer to equation (14) and (15), the single dot represents the first derivative and the double dot represents the second derivative and the three dots represent the third derivative. The equation (1) can be written as follows, y ⃛ ( t ) + 18 y ¨ ( t ) + 20 y ˙ ( t ) + 5 y ( t ) = z ( t ) (16) Substitute x ˙ 3 for y ⃛ ( t ) , x 3 for y ¨ ( t ) , x 2 for y ˙ ( t ) , and x 1 for y ( t ) in equation (16)

EE 98: Fundamentals of Electrical Circuits - With Connect Access

6th Edition

ISBN: 9781259981807

Author: Alexander

Publisher: MCG

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Chapter 16.5, Problem 12PP

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Develop a set of state variable equation for the given differential equation.

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Chapter 16 Solutions

EE 98: Fundamentals of Electrical Circuits - With Connect Access

Ch. 16.2 - Determine vo(t) in the circuit of Fig. 16.6,...Ch. 16.2 - Prob. 2PP Ch. 16.2 - Prob. 3PP Ch. 16.3 - For the circuit shown in Fig. 16.12 with the same...Ch. 16.3 - Prob. 5PP Ch. 16.3 - The initial energy in the circuit of Fig. 16.17 is...Ch. 16.4 - Prob. 7PP Ch. 16.4 - Prob. 8PP Ch. 16.4 - Prob. 9PP Ch. 16.5 - Obtain the state variable model for the circuit...

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Develop a set of state variable equation for the given differential equation.

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Develop a set of state variable equation for the given differential equation.

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Chapter 16 Solutions