To find a state-space realization of the transfer function

$G(s) = \frac{s^3}{s^3 + 3s^2 + 2s + 1},$

we can follow these steps:

1. Identify the Numerator and Denominator:

   • Numerator: $b(s) = s^3$
   • Denominator: $a(s) = s^3 + 3s^2 + 2s + 1$

2. Determine the Order: The order of the system is determined by the highest power of $s$ in the denominator, which is 3.

3. State-Space Representation: A typical state-space representation can be constructed in the following canonical form:

   \dot{x}(t) & = Ax(t) + Bu(t), \\ y(t) & = Cx(t) + Du(t), \end{align*}$$ where: - $$x(t)$$ is the state vector, - $$u(t)$$ is the input, - $$y(t)$$ is the output.

4. Construct the Matrices: The matrices $A$, $B$, $C$, and $D$ can be derived from the coefficients of the denominator and numerator:

   • The matrix $A$ will be a companion matrix formed from the coefficients of the denominator:
   0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & -3 \end{bmatrix}$$ - The input matrix $$B$$ will have a size matching the state vector: $$B = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$$ - The output matrix $$C$$ will be derived from the numerator: $$C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$$ - The direct transmission matrix $$D$$ is zero since there is no direct feedthrough from $$u$$ to $$y$$: $$D = 0$$

5. Final State-Space Representation: Thus, the state-space realization is:

\dot{x}(t) & = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & -3 \end{bmatrix} x(t) + \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} u(t), \\ y(t) & = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} x(t) + 0 \cdot u(t). \end{align*}$$ ### Summary The state-space realization for the given transfer function $$G(s)$$ is: - **A**: $$\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & -3 \end{bmatrix}$$ - **B**: $$\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$$ - **C**: $$\begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$$ - **D**: $$0$$ Would you like more details on any specific part of this process, or do you have any questions? Here are some related questions for further exploration: 1. What are the different forms of state-space realizations? 2. How does the controllability of this system affect its state-space representation? 3. What is the significance of the matrices $$A$$, $$B$$, $$C$$, and $$D$$? 4. How can you convert from state-space to transfer function form? 5. What are some applications of state-space representations in control systems? **Tip**: Familiarize yourself with different canonical forms of state-space realizations (controllable, observable, etc.) to enhance your understanding of system dynamics.