Problem 3 Suppose the following time invariant system is controllable and ob- servable. Prove that if it is BIBO stable, then it is asymptotically stable. ¿(t) = Aæ(t) + Bu(t) y(t) = Cx(t)
Problem 3 Suppose the following time invariant system is controllable and ob- servable. Prove that if it is BIBO stable, then it is asymptotically stable. ¿(t) = Aæ(t) + Bu(t) y(t) = Cx(t)
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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![**Problem 3**
Suppose the following time-invariant system is controllable and observable. Prove that if it is BIBO stable, then it is asymptotically stable.
\[
\dot{x}(t) = Ax(t) + Bu(t)
\]
\[
y(t) = Cx(t)
\]
In this problem, you are asked to analyze a time-invariant linear system described by the state-space equations. The system is defined as:
1. The state equation \(\dot{x}(t) = Ax(t) + Bu(t)\), where:
- \(\dot{x}(t)\) is the derivative of the state vector \(x(t)\) with respect to time.
- \(A\) is the system matrix that relates the current state \(x(t)\) to its rate of change.
- \(B\) is the input matrix that relates the input \(u(t)\) to the state.
2. The output equation \(y(t) = Cx(t)\), where:
- \(y(t)\) is the output vector.
- \(C\) is the output matrix that relates the state \(x(t)\) to the output.
Your task is to prove that if this system is BIBO (Bounded Input, Bounded Output) stable, given that it's both controllable and observable, then it is also asymptotically stable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9947224-5fba-4488-8402-388e5c0c3093%2Fc8dd4242-d454-4f17-87bc-e0521dd2101a%2Fqbwxqe_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3**
Suppose the following time-invariant system is controllable and observable. Prove that if it is BIBO stable, then it is asymptotically stable.
\[
\dot{x}(t) = Ax(t) + Bu(t)
\]
\[
y(t) = Cx(t)
\]
In this problem, you are asked to analyze a time-invariant linear system described by the state-space equations. The system is defined as:
1. The state equation \(\dot{x}(t) = Ax(t) + Bu(t)\), where:
- \(\dot{x}(t)\) is the derivative of the state vector \(x(t)\) with respect to time.
- \(A\) is the system matrix that relates the current state \(x(t)\) to its rate of change.
- \(B\) is the input matrix that relates the input \(u(t)\) to the state.
2. The output equation \(y(t) = Cx(t)\), where:
- \(y(t)\) is the output vector.
- \(C\) is the output matrix that relates the state \(x(t)\) to the output.
Your task is to prove that if this system is BIBO (Bounded Input, Bounded Output) stable, given that it's both controllable and observable, then it is also asymptotically stable.
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