1. Routh-Hurwitz Method

• method for determining the number of roots of a given polynomial that have positive real parts.
• Procedure: build the Routh-Hurwitz array, and then count the number of sign changes that occur in reading the first column from top to bottom.
• Applications:

1. Used for establishing the stability of a transfer function, if the polynomial being considered is the characteristic polynomial (i.e. denominator polynomial). Specifically, in cases where a design or control tuning parameter appear in the coefficients of the polynomial, the Routh Hurwitz can detect the acceptable ranges of these parameters (or even nonexistence of values) for which the system is stable.
2. Could also be used to detect the critical values of the parameters which could yield pure imaginary roots, as may be needed in a Ziegler-Nichols tuning approach.

2. Final Value Theorem.
 

Given L[x] = f(s), the value of  x at t=¥ can be found without having to invert f(s) first, via the following fact :

provided all the characteristic roots (i.e. roots of the denominator of L[x]) have negative real parts.

 

3. Frequency Response Methods

-Objective: to obtain dynamic relationship between the input and the output of a system using
    sinusoidal input ( black box modeling ) of different frequencies.

-Motivation:

The system may be too complex to model.

The frequency domain provides a powerful stability criterion and tools for robust design and filter designs.

- Experiment:

1. Introduce a sinusoidal input :

u = A sin (w t) if w is in rad/sec

or

u = A sin(2pwt) if w is in cycles/sec or Hz

2. Observe the output variable after it attains periodic behavior, i.e.

    y = B sin(w t + f) if w is in rad/sec

         or

         y = B sin(2pwt + f) if w is in cycles/sec or Hz

3. Perform the experiment for a range of w values and record the corresponding values of amplitude, B and time shift, t_shift( <0  if shifted to right ) and calculate

a) amplitude ratio,  AR = B/A
b) phase shift,  f = (t_shift/P)*360^o  (degrees)  = (t_shift/P)*2p  (radians)  =t_shift*w(radians)

         ( Note:   P=1/w if w is in cycles/sec or P=2p/w if w is in rads/sec )

       - Representations:

a) Bode Plots:

(1) LM (in decibels)    vs   w ( in log10 scale)
(2) f (in degrees)   vs   w (  in log10 scale)

                       where LM = log modulus = 20 log10 (B/A)

b) Nyquist Plots

B/A cos(f) in x-axis
B/A sin(f) in y-axis