Control System Stability - Homework - MATLAB

Discussion Question

Given the denominator of a closed-loop transfer function, ??⁴+ 20????³+ ??₁??²+ 4?? + ??₂= 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)  2??⁴  + ??³  + 3??²  + 5?? + 10

(b) ??³  + 3408.3??²  + 1,204,000?? + 1.5 × 10⁷??

(c)  ??³  + 3????²  + (?? + 2)?? + 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 ??_?? (??) = (??+??)(??+2)/?? (this is called a PID controller - we will cover PID controllers in Module 7) and the rocket transfer function ??(??) = ??/??²-1. 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.