ELE 317 Control Systems LaboratoryELE 327 Control Systems Laboratory

Laboratory: V

Title: Finding The Gain Value K At An Arbitrary Point on the Root Locus

Objectives: To have each student become familiar with MATLAB by engaging in exercises with concepts from sketching Root Locus and determining the Gain (K) from these plots.

In the MATLAB analysis of closed-loop systems, it is frequently desired to find the gain value K at an arbitrary point on the root locus. This can be accomplished by using either of the following rlocfind commands:

[K,r] = rlocfind(num,den) or [K,r] = rlocfind(A,B,C,D)

Here, r gives the closed-loop poles. The rlocfind command, which must follow an rlocus command, overlays moveable x-y coordinates on the screen. Using the mouse, we position the origin of the x-y coordinates over the desired point on the root locus and press the mouse button. Then MATLAB displays the coordinates of that point, the gain values at that point, and the closed-loop poles corresponding to the aforesaid gain value.

If the selected point is not on the root locus, the rlocfind command gives the coordinates of that selected point, the gain value of the point, and the locations of the closed-loop poles corresponding to that K value (Note every point on the s plane has a gain value.)

Example:

Consider the unity feedback control system with the following feedforward transfer function:

G(s) =

Plot the root loci with MATLAB. Determine the closed-loop poles that have the damping ratio 0.5. Find the gain value K at that point.

Example program:

>> num = [1]; >> den = [1 4 5 0]; >> K = 0:0.01:1000; >> r = rlocus(num,den); >> plot(r,-); v =[-3 1 -2 2]; axis(v); axis(square) >> grid >> sgrid(0.5,[]) >> [K,r] = rlocfind(num,den) >>xlabel(Real Axis); ylabel(Imag Axis)

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