Explanation Given data: The differential equation is, d 2 y ( t ) d t 2 + 7 d y ( t ) d t + 9 y ( t ) = d z ( t ) d 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: Assume the function, x 1 = y ( t ) (8) Take the first derivative for equation (8) to find x ˙ 1 . x ˙ 1 = y ˙ ( t ) (9) Assume, x ˙ 1 − z = x 2 Simplify the above equation to find x ˙ 1 . x ˙ 1 = x 2 + z (10) Substitute x 2 + z for x ˙ 1 in equation (9) to find y ˙ ( t ) . y ˙ ( t ) = x 2 + z (11) Take the first derivative for equation (11) to find y ¨ ( t ) . y ¨ ( t ) = x ˙ 2 + z ˙ Simplify the above equation to find x ˙ 2 . x ˙ 2 = y ¨ ( t ) − z ˙ (12) Refer to equation (12), the single dot represents the first derivative and the double dot represents the second derivative. The equation (1) can be written as follows, y ¨ ( t ) + 7 y ˙ ( t ) + 9 y ( t ) = z ˙ + z (13) Substitute x ˙ 2 + z ˙ for y ¨ ( t ) , x 2 + z for y ˙ ( t ) , and x 1 for y ( t ) in equation (13)

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 16, Problem 92P

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Develop the state equation for the given differential equation.

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Develop the state equation for the given differential equation.

Solution Summary: The author develops the state equation for the given differential equation. stackreldotx=Ax+Bz.

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