5 Questions
In the context of state variable technique, what does the transition matrix represent?
The state vector
What is the representation of the zero-input response in a linear system?
x(t) = Φ(t)x(0) + ∫Φ(t - τ)Bu(τ)dτ
What does the expression 'sX(s) - x(0) = AX(s) + BU(s)' represent in the context of linear systems?
Zero-state response
What is the function of the transition matrix in relation to the state variable technique?
Relates the state and input vectors
In a multivariable system, what does the state vector represent?
The internal state of the system
Study Notes
State Transition Matrix and Homogeneous Systems
- A linear time-invariant system is represented by the equations: dx(t)/dt = Ax(t) + Bu(t) and y(t) = Cx(t) + Du(t)
- The behavior of x(t) and y(t) can be analyzed in two parts: homogeneous solution and non-homogeneous solution
- A homogeneous system is one whose properties are either the same throughout the system or vary continuously from point to point with no discontinuities
- The principle of homogeneity states that if a system generates an output y(t) for an input x(t), it must produce an output ay(t) for an input ax(t)
Homogeneous Solution
- The homogeneous solution x(t) is the solution to the equation x˙(t) = Ax(t)
- The homogeneous solution is given by x(t) = e^At x(0)
- The state transition matrix Φ(t) is defined as Φ(t) = e^At
- The state transition matrix can be used to find the state x(t) at any time t, given the initial state x(0) and the time elapsed t
Properties of State Transition Matrix
- Φ(0) = I (identity matrix)
- Φ(t1 + t2) = Φ(t1) Φ(t2)
- Φ(-t) = [Φ(t)]^-1
- Φ(t) is non-singular and hence invertible
Test your understanding of power systems dynamics equations and their solutions. Determine the behavior of x(t) and y(t), and grasp the principle of homogeneity in system responses.
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