## 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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