when calculating the linear matrix for constructing the LQR controller, from fractional Van der pol system we can get # the matrix for the linear part of the system dynamics

A = np.array([[0, 1], [-1, a]]), so then for the fractional Rossler system, what should the matrix A be like?

The matrix \( A \) used in the construction of an LQR (Linear-Quadratic Regulator) controller corresponds to the linearized system dynamics around an equilibrium point for the system. The matrix \( A \) represents the Jacobian matrix of the system's equations, evaluated at that equilibrium point.

For the fractional Rössler system, the system equations are:

\[
\begin{aligned}
D^\alpha x &= -y - z \\
D^\alpha y &= x + a y \\
D^\alpha z &= b + z(x - c)
\end{aligned}
\]

To construct the matrix \( A \), we first linearize the system around an equilibrium point \( (x_0, y_0, z_0) \). Usually, this is the origin \( (0, 0, 0) \) for simplicity. The linearized system is obtained by taking the partial derivatives of each equation with respect to the state variables \( x \), \( y \), and \( z \).

### Partial Derivatives

Let's calculate the partial derivatives for the linearization:

1. **First equation**: \( D^\alpha x = -y - z \)
   - \(\frac{\partial}{\partial x} (-y - z) = 0 \)
   - \(\frac{\partial}{\partial y} (-y - z) = -1 \)
   - \(\frac{\partial}{\partial z} (-y - z) = -1 \)

2. **Second equation**: \( D^\alpha y = x + a y \)
   - \(\frac{\partial}{\partial x} (x + a y) = 1 \)
   - \(\frac{\partial}{\partial y} (x + a y) = a \)
   - \(\frac{\partial}{\partial z} (x + a y) = 0 \)

3. **Third equation**: \( D^\alpha z = b + z(x - c) \)
   - \(\frac{\partial}{\partial x} (b + z(x - c)) = z \) (Evaluating at the origin \( z = 0 \))
   - \(\frac{\partial}{\partial y} (b + z(x - c)) = 0 \)
   - \(\frac{\partial}{\partial z} (b + z(x - c)) = x - c \) (Evaluating at the origin \( x = 0 \))

### Jacobian Matrix \( A \)

After calculating the partial derivatives and evaluating them at the equilibrium point \( (x_0, y_0, z_0) = (0, 0, 0) \), the Jacobian matrix \( A \) is:

\[
A = \begin{bmatrix}
\frac{\partial (-y - z)}{\partial x} & \frac{\partial (-y - z)}{\partial y} & \frac{\partial (-y - z)}{\partial z} \\
\frac{\partial (x + a y)}{\partial x} & \frac{\partial (x + a y)}{\partial y} & \frac{\partial (x + a y)}{\partial z} \\
\frac{\partial (b + z(x - c))}{\partial x} & \frac{\partial (b + z(x - c))}{\partial y} & \frac{\partial (b + z(x - c))}{\partial z}
\end{bmatrix}
=
\begin{bmatrix}
0 & -1 & -1 \\
1 & a & 0 \\
0 & 0 & -c
\end{bmatrix}
\]

So, for the fractional Rössler system, the linearized system dynamics matrix \( A \) around the origin is:

\[
A = \begin{bmatrix}
0 & -1 & -1 \\
1 & a & 0 \\
0 & 0 & -c
\end{bmatrix}
\]

This matrix \( A \) can then be used in the LQR controller design process.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How does the LQR controller work in the context of a fractional-order system?
2. Can you explain how to calculate the controllability and observability matrices for the fractional Rössler system?
3. What are the effects of the fractional order \( \alpha \) on the system's stability?
4. How do you choose the weighting matrices \( Q \) and \( R \) in the LQR design?
5. What are some numerical methods for solving fractional-order linear systems?

**Tip:** When working with fractional-order systems, it's important to verify that your linearization is valid around the chosen equilibrium point by checking the system's behavior in that region.

Learn how to construct the matrix A for the fractional Rössler system to design an LQR controller. This detailed explanation covers linearization, Jacobian matrix calculation, and its application in controlling fractional-order systems.

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