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Questions and Answers
In the context of state variable technique, what does the transition matrix represent?
In the context of state variable technique, what does the transition matrix represent?
- The input vector
- The state vector (correct)
- The output vector
- The system matrix
What is the representation of the zero-input response in a linear system?
What is the representation of the zero-input response in a linear system?
- y(t) = C(t)x(t) + D(u)
- y(t) = CΦ(t - t0)x(t0) + ∫CΦ(t - τ)Bu(τ)dτ + Du(t)
- dx = A(t)x(t) + B(t)u(t)
- x(t) = Φ(t)x(0) + ∫Φ(t - τ)Bu(τ)dτ (correct)
What does the expression 'sX(s) - x(0) = AX(s) + BU(s)' represent in the context of linear systems?
What does the expression 'sX(s) - x(0) = AX(s) + BU(s)' represent in the context of linear systems?
- Zero-state response (correct)
- Zero-input response
- State variable representation
- Non-homogeneous solution
What is the function of the transition matrix in relation to the state variable technique?
What is the function of the transition matrix in relation to the state variable technique?
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In a multivariable system, what does the state vector represent?
In a multivariable system, what does the state vector represent?
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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 xË™(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
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