Consider the following feedback system, where P(s) = s + 4/s^2 + 2s + 2 For each

Question

Consider the following feedback system, where P(s) = s + 4/s^2 + 2s + 2 For each design task below, synthesize a PID controller that places closed-loop poles (not necessarily dominant) at s_1, 2 = - 3 plusminus j1. Additionally, for each part, use Matlab to produce two plots: a root locus graph and a closed-loop step response (demonstrating that your controller takes the steady-state error to zero). Include your Matlab code in your solution. Design a controller of the form C(s) = K(s + alpha) (s + alpha_2)/s. Design a controller of the form C(s) = K (s + alpha)^2/s.

Explanation / Answer

Problem #3

The above figure can be written as sys=C*P.............(1)

If sys is a transfer function then,

h(s)=n(s)*d(s)...............(2)

the closed-loop poles are the roots of

d(s)+kn(s)=0

rlocus selects a set of positive gains k to produce a smooth plot. Also,

uses the user-specified vector k of gains to plot the root locus.

rlocus(sys1,sys2,...) draws the root loci of multiple LTI models sys1, sys2,... on a single plot.

[r,k] = rlocus(sys) and r = rlocus(sys,k) return the vector k of selected gains and the complex root locations are for these gains. The matrix r has length(k) columns and its jth column enlists the closed-loop roots for the gain k(j).

The code for root locus is as follows:

The below is the code for closed loop step response:

PID synthesis is dependent on conditions.

One important criterion for control design is the selection of the closed-loop poles sufficiently far from the imaginary-axis of the complex-plane to have small time-constants, implying short settling times. Therefore, it is desirable for the closed-loop poles to have real-parts less than h for a pre-specified positive constant h. To identify plant classes such that closed-loop poles can be assigned to the left of an axis shifted away from the origin and to develop a synthesis procedure that describes PID controllers can be achived.

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