Explanation Given data: The differential equation is, d 3 y ( t ) d t 3 + 6 d 2 y ( t ) d t 2 + 11 d y ( t ) d t + 6 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). 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 ) + 6 y ¨ ( t ) + 11 y ˙ ( t ) + 6 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)

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ISBN: 9781259663918

Author: Alexander

Publisher: MCG

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Chapter 16, Problem 93P

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Develop the state equation for the given differential equation.

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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.

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