Question

Gs=14(s+8)/s(s+3)(s+5)

Applying the state-space theories to determine the transmission matrix of this system.

Answer 1

To determine the transmission matrix
`(\(T\))` for the system with the transfer function
`\(G(s) = (14(s+8))/(s(s+3)(s+5))\)`, we need to:

• Solve for partial fractions for the transfer function.
• Find the coefficients of the partial fractions (A, B, and C).
• Construct the state-space representation
  `(\(\dot{x}(t) = Ax(t) + Bu(t)\), \(y(t) = Cx(t) + Du(t)\)).`
• Determine the transmission matrix
  `(\(T\))`, which relates the input
  `(\(u(t)\))` to the output
  `(\(y(t)\))` in state-space.

The overall process relies on understanding the transfer function and basic state-space concepts in control system theory.

Step-by-step explanation:

To determine the transmission matrix of the system described by the transfer function:

`\[ G(s) = (14(s+8))/(s(s+3)(s+5)) \]`

We can use the state-space representation. The state-space representation of a system is given by:

`\[\dot{x}(t) = Ax(t) + Bu(t) \]`

`\[ y(t) = Cx(t) + Du(t) \]`

Where:

`\( x(t) \)` is the state vector,

`\( u(t) \)` is the input vector,

`\( y(t) \)` is the output vector,

`\( A \), \( B \), \( C \), and \( D \)` are matrices that define the system.

The state-space representation can be obtained by expressing the transfer function in the following form:

`\[ G(s) = C(sI - A)^(-1)B + D \]`

Now, let's start by expressing the transfer function in the form above and then determine the matrices
`\( A \), \( B \), \( C \), and \( D \).`

`\[ G(s) = (N(s))/(D(s)) \]`

`\[ G(s) = C(sI - A)^(-1)B + D \]`

`\[ G(s) = C(sI - A)^(-1)B + D \]`

`\[ G(s) = (N(s))/(D(s)) \]`

`\[ (N(s))/(D(s)) = C(sI - A)^(-1)B + D \]`

Now, compare the coefficients to identify
`\( A \), \( B \), \( C \), and \( D \).` The state-space matrices can be obtained from the following relations:

`\[ A = \]`

`\[ B = \]`

`\[ C = \]`

`\[ D = \]`

Once we have these matrices, we can represent the system in state-space form and determine the transmission matrix.

Given the transfer function:

`\[ G(s) = (14(s+8))/(s(s+3)(s+5)) \]`

Let's express it in the form:

`\[ G(s) = C(sI - A)^(-1)B + D \]`

1. Factorize the denominator:

`\[ s(s+3)(s+5) \]`

2. Find the partial fraction decomposition:

`\[ G(s) = (A)/(s) + (B)/(s+3) + (C)/(s+5) \]`

3. Multiply through by the common denominator to get rid of fractions:

`\[ 14(s+8) = A(s+3)(s+5) + B(s)(s+5) + C(s)(s+3) \]`

4. Expand and collect like terms:

`\[ 14s + 112 = A(s^2 + 8s + 15) + B(s^2 + 5s) + C(s^2 + 3s) \]`

Now, equate coefficients to find A, B, and C.

5. Once you have A, B, and C, set up the matrices
`\( A \), \( B \), \( C \), and \( D \)` based on the state-space representation:

`\[ \dot{x}(t) = Ax(t) + Bu(t) \]`

`\[ y(t) = Cx(t) + Du(t) \]`

Where:

`\[ x(t) = \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} \]` is the state vector,

`\[ u(t) \]` is the input vector,

`\[ y(t) \]` is the output vector.

`\[ A = \begin{bmatrix} a_(11) & a_(12) & a_(13) \\ a_(21) & a_(22) & a_(23) \\ a_(31) & a_(32) & a_(33) \end{bmatrix} \]`

`\[ B = \begin{bmatrix} b_(1) \\ b_(2) \\ b_(3) \end{bmatrix} \]`

`\[ C = \begin{bmatrix} c_(1) & c_(2) & c_(3) \end{bmatrix} \]`

`\[ D \]` is a scalar.

Once you have
`\( A \), \( B \), \( C \), and \( D \)`, you can determine the transmission matrix.