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The state transition matrix for the system X- = AX with initial state X(0) is
- a)(sI-A)-1
- b)eA tX(0)
- c)Laplace inverse of [(s I-A)-1]
- d)Laplace inverse of [(sI-A)-1X (0)]
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The state transition matrix for the system X- = AX with initial state ...
Correct option is C.
Laplace inverse of [(sI−A)−1]
eAt = L−1[sI−A]−1
Laplace inverse of [(sI−A)−1]
eAt = L−1[sI−A]−1
Most Upvoted Answer
The state transition matrix for the system X- = AX with initial state ...
Explanation:
To understand why the correct answer is option 'C', let's break down the given options and discuss each one.
Option A: (sI-A)-1
This represents the inverse Laplace transform of (sI-A)-1. However, the Laplace transform is used to solve initial value problems, not state space equations. Therefore, this option is not applicable in this case.
Option B: eAtX(0)
This represents the solution to the state space equation X- = AX with initial condition X(0). It is the form of the solution when exponential matrix is used. However, this option is not the state transition matrix.
Option C: Laplace inverse of [(s I-A)-1]
This is the correct answer. The Laplace inverse of [(s I-A)-1] gives the state transition matrix. The state transition matrix, denoted as ϕ(t), is defined as the solution to the state space equation X- = AX with initial condition X(0) = I, where I is the identity matrix. The state transition matrix represents the time evolution of the system.
Option D: Laplace inverse of [(sI-A)-1X(0)]
This option is incorrect because it represents the solution to the state space equation X- = AX with initial condition X(0). It is the form of the solution when the initial condition is given. However, the state transition matrix is independent of the initial condition.
Conclusion:
The correct answer is option 'C' because the Laplace inverse of [(s I-A)-1] gives the state transition matrix, which represents the time evolution of the system.
To understand why the correct answer is option 'C', let's break down the given options and discuss each one.
Option A: (sI-A)-1
This represents the inverse Laplace transform of (sI-A)-1. However, the Laplace transform is used to solve initial value problems, not state space equations. Therefore, this option is not applicable in this case.
Option B: eAtX(0)
This represents the solution to the state space equation X- = AX with initial condition X(0). It is the form of the solution when exponential matrix is used. However, this option is not the state transition matrix.
Option C: Laplace inverse of [(s I-A)-1]
This is the correct answer. The Laplace inverse of [(s I-A)-1] gives the state transition matrix. The state transition matrix, denoted as ϕ(t), is defined as the solution to the state space equation X- = AX with initial condition X(0) = I, where I is the identity matrix. The state transition matrix represents the time evolution of the system.
Option D: Laplace inverse of [(sI-A)-1X(0)]
This option is incorrect because it represents the solution to the state space equation X- = AX with initial condition X(0). It is the form of the solution when the initial condition is given. However, the state transition matrix is independent of the initial condition.
Conclusion:
The correct answer is option 'C' because the Laplace inverse of [(s I-A)-1] gives the state transition matrix, which represents the time evolution of the system.
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The state transition matrix for the system X- = AX with initial state X(0) isa)(sI-A)-1b)eA tX(0)c)Laplace inverse of [(s I-A)-1]d)Laplace inverse of [(sI-A)-1X (0)]Correct answer is option 'C'. Can you explain this answer?
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The state transition matrix for the system X- = AX with initial state X(0) isa)(sI-A)-1b)eA tX(0)c)Laplace inverse of [(s I-A)-1]d)Laplace inverse of [(sI-A)-1X (0)]Correct answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about The state transition matrix for the system X- = AX with initial state X(0) isa)(sI-A)-1b)eA tX(0)c)Laplace inverse of [(s I-A)-1]d)Laplace inverse of [(sI-A)-1X (0)]Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The state transition matrix for the system X- = AX with initial state X(0) isa)(sI-A)-1b)eA tX(0)c)Laplace inverse of [(s I-A)-1]d)Laplace inverse of [(sI-A)-1X (0)]Correct answer is option 'C'. Can you explain this answer?.
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