Represent the following transfer functions in state-space representation. Give your answer in vector- matrix form. a. 2s² + s +3 s4 +8s³ - 3s² + 32s + 40 (s+2)(s+4) (s + 3) (s² + 1) +4
Transfer function
A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.
Convolution Integral
Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.
![**Title: State-Space Representation of Transfer Functions**
**Objective:**
Convert the given transfer functions into state-space representation. The answer should be provided in vector-matrix form.
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**Problem Statement:**
Represent the following transfer functions in state-space representation:
### a.
\[
\frac{2s^2 + s + 3}{s^4 + 8s^3 - 3s^2 + 32s + 40}
\]
### b.
\[
\frac{(s + 2)(s + 4)}{(s + 3)(s^2 + 1)} + 4
\]
**Instructions:**
1. Express each transfer function using state-space representation by defining:
- State matrix \( A \)
- Input matrix \( B \)
- Output matrix \( C \)
- Feedthrough (direct transmission) matrix \( D \)
2. Ensure the matrices accurately reflect the dynamics of the given system as described by each transfer function, facilitating analysis or controller design.
**Graphical/Diagram Explanation:**
- No graphs or diagrams are included with the problem. Ensure any resulting matrices are neatly organized to maintain clarity when transitioned to educational platforms.
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**Note:** The correct transformation into state-space form requires familiarity with concepts such as controllable or observable canonical forms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e11f595-a40a-4bf3-b7d2-0a0cab1f1556%2F79c5ee74-03cb-4cbe-914a-523cf5e4adf8%2F9sqch99_processed.jpeg&w=3840&q=75)
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