15 Questions
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
In the context of a linear system, what does $\Phi(t)$ represent?
Transition matrix
What does the equation $y(t) = C(t)x(t) + D(u)$ represent in a linear system?
Output vector
What is the role of matrix $B$ in the state-space model equation $dx = A(t)x(t) + B(t)u(t)dt$?
To describe the input vector
What does the term 'Zero-state response' refer to in the context of a linear system?
The response of a system with zero initial conditions
What is the dual concept of controllability?
Observability
How is the observability of an input-free system studied?
By obtaining derivatives of continuous-time measurements
What criterion ensures a unique solution for a system of linear algebraic equations with n unknowns?
The system matrix has rank n
How is the completeness of the rank of square matrices tested?
By finding their determinants
Under what condition is a linear continuous-time system considered controllable?
The controllability matrix C has full rank
What is the dual concept of controllability?
Observability
How is the observability of an input-free system studied?
By taking derivatives of the continuous-time measurements
Under what condition is a linear continuous-time system considered controllable?
If the controllability matrix has full rank
What criterion ensures a unique solution for a system of linear algebraic equations with n unknowns?
Full rank of the system matrix
What is the role of matrix $B$ in the state-space model equation $dx = A(t)x(t) + B(t)u(t)dt$?
Representing an input in the state-space model
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
Test your understanding of state transition matrix and the behavior of x(t) and y(t) in power systems. Explore concepts such as homogeneous and non-homogeneous solutions as well as the principle of homogeneity.
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