Understanding State-Space Representation
State-space representation is a mathematical model that describes a system using a set of input, output, and state variables. It is advantageous due to its ability to handle multiple-input and multiple-output (MIMO) systems and non-linear dynamics.
Transfer Function Derivation
To derive the transfer function \( C(s)/R(s) \) from state-space equations, we start with the standard state-space form:
- State Equation:
\( \dot{x}(t) = Ax(t) + Bu(t) \)
- Output Equation:
\( y(t) = Cx(t) + Du(t) \)
Where:
- \( x(t) \) = state vector
- \( u(t) \) = input vector
- \( y(t) \) = output vector
- \( A \), \( B \), \( C \), and \( D \) = matrices defining the system dynamics
Transforming to the Frequency Domain
To find the transfer function, apply the Laplace Transform to the state and output equations:
- \( sX(s) - x(0) = AX(s) + BU(s) \)
- \( Y(s) = CX(s) + DU(s) \)
Where \( X(s) \), \( U(s) \), and \( Y(s) \) are the Laplace Transforms of the state, input, and output respectively.
Solving for Transfer Function
Rearranging the state equation gives:
- \( (sI - A)X(s) = x(0) + BU(s) \)
Solving for \( X(s) \):
- \( X(s) = (sI - A)^{-1}(x(0) + BU(s)) \)
Substituting \( X(s) \) into the output equation yields:
- \( Y(s) = C(sI - A)^{-1}(x(0) + BU(s)) + DU(s) \)
Assuming zero initial conditions (\( x(0) = 0 \)) simplifies this to:
- \( Y(s) = C(sI - A)^{-1}B U(s) + DU(s) \)
Thus, the transfer function \( H(s) = \frac{Y(s)}{U(s)} \) is given by:
- Transfer Function:
\( H(s) = C(sI - A)^{-1}B + D \)
This matrix formulation encapsulates the dynamics and behavior of the system, providing a comprehensive view of its response to inputs.