ControlsHelpers

My HP Prime Helper Functions for Control Systems Analysis

Directions

1. Update your firmware.
2. Drag the *.hpprgm into Programs directory.
3. Rename the program by deleting the "Local" word in the filename.

Usage

SecondOrderSystem

• DampingRatio(os) calculates the damping ratio zeta for a given overshoot percentage (expressed as a fraction of 1)
• Overshoot(zeta) calculates the overshoot (expressed as a fraction of 1) given a damping ratio
• PeakTime(zeta,omega) calculates the peak time of a second-order system given a zeta and omega input.
• RiseTime(zeta,omega) calculates the rise time of a second-order system given a zeta and omega input. Note the result is a third-order polynomial estimate due to the absence of a closed-form solution.
• SettlingTime(zeta,omega) calculates the settling time of a second-order system given a zeta and omega input.
• PeakMagnitude(zeta) calculates the peak magnitude of a second-order open-loop system given zeta input
• PeakFrequency(zeta,omega) calcaultes the frequency at the peak magnitude of a second-order open-loop system given zeta and omega input
• Bandwidth(zeta,omega) calculates the bandwidth of a second-order open-loop system given zeta and omega.
• PhaseMargin(zeta) calculates the phase margin of a second-order open-loop system given zeta.

RootLocus

• rldamp(numerator, denominator, zeta, radius) calculates the total angle contributions of a critical point for an open-loop transfer function with polynomial coefficients numerator and denominator to a point given by damping ratio zeta and polar radius radius. This program is mainly used to check whether a given point in the complex plane lies on the root locus. The inputs numerator and denominator are row matrices consisting of the coefficients of the relevant polynomials in decreasing power, so [1 2 3] corresponds to the coefficients of the polynomial x^2 + 2x + 3.

• rlk(numerator, denominator, complex_point) calculates the required gain for the open-loop transfer function with polynomial coefficients numerator and denominator to realize a pole of complex_point, expressed as a+b*i. For example, after using rldamp to confirm the exist of a pole, its coordinates can be converted to Cartesian via:

  radius*cos(180-acos(zeta)) + radius*sin(180-acos(zeta))*i
  

• rlbreak(numerator, denominator) calculates breakaway and break-in points for the for an open-loop transfer function with polynomial coefficients numerator and denominator. It accomplishes this by solving for the zeros of dK/ds = d(-P(s)/Q(s))/ds, where T(s) = Q(s)/P(s).

Frequency Response

• ReImParts(expression) generates a list where the first entry is the real component and second entry is the imaginary component of any expression.
• Mag(expression) calculates the magnitude of a frequency response function
• Phase(expression) calculates the phase of a frequency response function
• FreqResponseEval(num, den, omega) evaluates the frequency response G(j*omega) of a transfer function with num,den being the coefficients of its numerator and denominator respectively
• MagContrib(num, den, c) evaluates the magnitude of a frequency response given a complex point c and transfer function coefficients
• PhaseContrib(num, den, c) evaluates the phase of a frequency response given a complex point c and transfer function coefficients

Third Party Software

The following are mirrors of applications from the HP Developer website:

• Control Systems Application
• Routh Hurwitz

Root Locus - Integrated Example

Consider the unity feedback loop with forward transfer function

        K(s+1)(s+2)
G(s) = ------------
        (s-1)(s-2)

Calculate the value of K that yields a stable system with a pair of second-order poles that have a damping ratio of 0.707. Where are the break-in and breakaway points?

To find the damping ratio of 0.707 and find the second-order poles. We can search the line at angle pi - acos(0.707) for different values of radius with the command

rldamp([1 3 2],[1 -3 2],0.707,sqrt(2)]
                                    -180.0

This implies a radius of sqrt(2) is needed for the overshoot line of 0.707 zeta to intersect the root locus. The associated gain is

rlk([1 3 2],[1 -32],sqrt(2)*cos(180-acos(0.707))+sqrt(2)*sin(180-acos(0.707))*i)
                                    4.9981

Finally, the breakaway and break-in points are

rlbreak([1 3 2],[1 -3 2])
                                    [1.41421356237 -1.41421356237]

Frequency Response - Integrated Example

Consider the unity feedback system with the forward transfer function

              6
G(s) = ---------------
       (s^2+2s+2)(s+2)

Find the gain and phase margin.

To find the gain margin, we first find where the Nyquist diagram crosses the real axis for frequency between 0 and positive infinity.

We can do this by first evaluating the frequency response with a variable, converting it to complex form, and setting the imaginary part equal to zero and solving the the real part. We get a real part of -0.3. Thus K can be increased 3.33 before the real part becomes -1, and the gain margin Gm = 20 log 3.33 = 10.45 dB.

The phase margin can be found by searching positive frequencies that give a magnitude of unity. This frequency of 1.253 rad/s is then used to calculate the total phase contribution, which is -112.3 degrees. This results in a phase margin of -180 - (-112.3) = 67.7 degrees.