Control System Stability - Homework - MATLAB

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Discussion Question

• Given the denominator of a closed-loop transfer function, 𝑆𝑆4 + 20𝑠𝑠3 + 𝐾𝐾1𝑆𝑆2 + 4𝑆𝑆 + 𝐾𝐾2 = 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) 2𝑠𝑠4 + 𝑠𝑠3 + 3𝑠𝑠2 + 5𝑠𝑠 + 10 (b) 𝑠𝑠3 + 3408.3𝑠𝑠2 + 1,204,000𝑠𝑠 + 1.5 × 107𝐾𝐾 (c) 𝑠𝑠3 + 3𝐾𝐾𝑠𝑠2 + (𝐾𝐾 + 2)𝑠𝑠 + 4

3. 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 𝐺𝐺𝑐𝑐(𝑠𝑠) = (𝑠𝑠+𝑚𝑚)(𝑠𝑠+2)

𝑠𝑠 (this is called a PID controller - we will cover PID

controllers in Module 7) and the rocket transfer function 𝐺𝐺(𝑠𝑠) = 𝐾𝐾 𝑠𝑠2−1

. 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.

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