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Questions and Answers
What is the purpose of the matrix $A$ in the equation $dx = A(t)x(t) + B(t)u(t)dt$?
What is the purpose of the matrix $A$ in the equation $dx = A(t)x(t) + B(t)u(t)dt$?
- To model the system dynamics (correct)
- To define the state vector
- To represent the input vector
- To describe the state transition
In the context of a linear system, what does $\Phi(t)$ represent?
In the context of a linear system, what does $\Phi(t)$ represent?
- State variable technique
- Zero-input response
- Transition matrix (correct)
- State representation of a linear system
What does the equation $y(t) = C(t)x(t) + D(u)$ represent in a linear system?
What does the equation $y(t) = C(t)x(t) + D(u)$ represent in a linear system?
- State transition matrix
- Output vector (correct)
- Input vector
- State variable technique
What is the role of matrix $B$ in the state-space model equation $dx = A(t)x(t) + B(t)u(t)dt$?
What is the role of matrix $B$ in the state-space model equation $dx = A(t)x(t) + B(t)u(t)dt$?
What does the term 'Zero-state response' refer to in the context of a linear system?
What does the term 'Zero-state response' refer to in the context of a linear system?
What is the dual concept of controllability?
What is the dual concept of controllability?
How is the observability of an input-free system studied?
How is the observability of an input-free system studied?
What criterion ensures a unique solution for a system of linear algebraic equations with n unknowns?
What criterion ensures a unique solution for a system of linear algebraic equations with n unknowns?
How is the completeness of the rank of square matrices tested?
How is the completeness of the rank of square matrices tested?
Under what condition is a linear continuous-time system considered controllable?
Under what condition is a linear continuous-time system considered controllable?
What is the dual concept of controllability?
What is the dual concept of controllability?
How is the observability of an input-free system studied?
How is the observability of an input-free system studied?
Under what condition is a linear continuous-time system considered controllable?
Under what condition is a linear continuous-time system considered controllable?
What criterion ensures a unique solution for a system of linear algebraic equations with n unknowns?
What criterion ensures a unique solution for a system of linear algebraic equations with n unknowns?
What is the role of matrix $B$ in the state-space model equation $dx = A(t)x(t) + B(t)u(t)dt$?
What is the role of matrix $B$ in the state-space model equation $dx = A(t)x(t) + B(t)u(t)dt$?
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Study Notes
Controllability and Observability
- Controllability and observability are two major concepts of modern control system theory
- Controllability: the system must be controllable to do whatever we want with the given dynamic system under control input
- Observability: the system must be observable to see what is going on inside the system under observation
Controllability
- Definition: the process G is said to be controllable if every state variable x of G can be affected or controlled in finite time by some unconstrained control signal u(t)
Observability
- Definition: the process G is said to be observable if every state variable x of G eventually affects some of the outputs y of the process
State Transition Matrix
- The state transition matrix is denoted by Φ(t) = e^(At)
- It satisfies the following properties:
- Φ(0) = I
- Φ(t1 + t2) = Φ(t1) Φ(t2)
- Φ(t) = Φ^(-1)(-t)
- Φ(t) is non-singular for all t
- Φ(t) is continuous for all t
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