Given The Denominator Of A Closed-Loop Transfer Function

Question

Given the denominator of a closed-loop transfer function, Given the denominator of a closed-loop transfer function, ð‘†ð‘†4 + 20ð‘ ð‘ 3 + ð¾ð¾1ð‘†ð‘†2 + 4ð‘†ð‘† + ð¾ð¾2 = 0, discuss what values of K1 and K2 will lead to a stable system. Solve the following problems: 1. For the given system below: Determine the range of K for the system to become unstable. 2. Determine the stability of the following polynomials: (a) 2ð‘ ð‘ 4 + ð‘ ð‘ 3 + 3ð‘ ð‘ 2 + 5ð‘ ð‘ + 10 (b) ð‘ ð‘ 3 + 3408.3ð‘ ð‘ 2 + 1,204,000ð‘ ð‘ + 1.5 à— 107ð¾ð¾ (c) ð‘ ð‘ 3 + 3ð¾ð¾ð‘ ð‘ 2 + (ð¾ð¾ + 2)ð‘ ð‘ + . for the following system: (a) Determine the range of K for stability. (b) Develop an m-file to calculate the closed-loop poles for K from 0 to 5 with an increment of 0.1 (you may want to use the for loop in MATLAB). What are the poles when K = 4? Design Project The altitude control of a rocket is shown in the following figure: The controller given is ðºðºð‘ð‘(ð‘ ð‘ ) = (ð‘ ð‘ +ð‘šð‘š)(ð‘ ð‘ +2) ð‘ ð‘ (this is called a PID controller - we will cover PID controllers in Module 7) and the rocket transfer function ðºðº(ð‘ ð‘ ) = ð¾ð¾ ð‘ ð‘ 2âˆ’1. Note that the rocket itself is open-loop unstable (a pole is on the right hand side of the complex plane) and feedback with a controller is needed to stabilize the system. 1. Using the Routh-Hurwitz criterion, determine the range of K and m so that the system is stable, and plot the region of stability (m vs. K). 2. Select K and m so that the steady-state error due to a ramp input is less or equal to 10% of the input magnitude. With K and m you selected from Part 2, write a MATLAB program to obtain and plot the unit step response of the system, and determine the percent overshoot (P.O.) of the system from your plot.

Dr. Jack HW Helper · Accepted Answer

Control systems and stability analysis form the backbone of electrical engineering and are crucial in various applications, particularly in rocket altitude control systems which exhibit complex dynamic behavior. This paper discusses a closed-loop transfer function denoted by the polynomial: $s^4 + 20s^3 + K_1 s^2 + 4s + K_2 = 0$, providing an in-depth analysis of values for $K_1$ and $K_2$ that ensure system stability, alongside additional polynomial stability evaluations and design project considerations for a rocket controller. Understanding Closed-Loop Transfer Functions The stability of a closed-loop control system can be determined by analyzing the poles of its characteristic equation. For the polynomial presented, elements $K_1$ and $K_2$ greatly influence system dynamics. A stable system requires that all poles have negative real parts, evident from the location of the poles in the complex plane. The Routh-Hurwitz criterion serves as a precise method for establishing the conditions needed for stability based on the $K_1$ and $K_2$ parameters. A stability analysis will yield the ranges of $K_1$ and $K_2$, indicating the necessary conditions for achieving the desired behavior of the control system. Evaluating System Instability We will first determine the range of $K$ for when the systems become unstable, specifically evaluating control polynomials: (a) $2s^4 + s^3 + 3s^2 + 5s + 10$ (b) $s^3 + 3408.3s^2 + 1,204,000s + 1.5 \times 10^7$ (c) $s^3 + 3 \varepsilon s^2 + (\eta + 2)s + ... $ For each polynomial, stability can be analyzed via the Routh-Hurwitz criterion, ensuring the array is filled and examining the sign changes. Details of the First Polynomial Assessing polynomial (a), coefficients are: $a_0 = 10$, $a_1 = 5$, $a_2 = 3$, $a_3 = 1$, and $a_4 = 2$. The construction of the Routh array starts with these coefficients yielding a first column that shows positivity, indicating potential stability depending upon the ranges o

Given The Denominator Of A Closed-Loop Transfer Function ✓ Solved

Given the denominator of a closed-loop transfer function,

Paper For Above Instructions

Understanding Closed-Loop Transfer Functions

Evaluating System Instability

Details of the First Polynomial

Determining Stability and Unstable Region for K

Designing a PID Controller for Rocket Altitude Control

Impact of Steady-State Errors

Conclusion

References