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)

EE 98: Fundamentals of Electrical Circuits - With Connect Access

6th Edition

ISBN: 9781259981807

Author: Alexander

Publisher: MCG

See similar textbooks

Concept explainers

Transfer function

A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.

Convolution Integral

Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is …

Question

Chapter 16, Problem 92P

To determine

Develop the state equation for the given differential equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 16, Problem 91P

Chapter 16, Problem 93P

Students have asked these similar questions

Pls show neat and whole solution

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...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, electrical-engineering and related others by exploring similar questions and additional content below.

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.

Concept explainers

Develop the state equation for the given differential equation.

Want to see the full answer?

Chapter 16 Solutions