AME-558

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Topic 1: Review of Classical Control Theory

• 1.1 Open and closed loop poles
• 1.2 Stability analysis
• 1.3 Steady state errors
• 1.4 Transient response
• 1.5 Frequency response

System Performance

Criteria in analysis of system performance are: stability, steady-state accuracy, transient response, frequency response, and robustness. Classical methods of analysis corresponding with these criteria are:

Stability

For now, we can define stability to mean:

Bounded input → Bounded output

The principal stability criterion for linear systems is:

A system is stable if all poles of its transfer function lie in the left-half of the complex s-plane.

Equivalently, a system is stable if the real parts of all roots of its characteristic equation are negative. The characteristic equation is found simply by equating the denominator of the transfer function to zero. Note that a root of the characteristic equation is synonymous with a system pole.

For a realizable feedback system, we expect the transfer function to be expressible in the following form:

$\frac{C\left(s\right)}{R\left(s\right)}=\frac{KG\left(s\right)}{1+KG\left(s\right)H\left(s\right)}=\frac{K{g}_{\text{n}}\left(s\right)h_{\text{d}}\left(s\right)}{g_{\text{d}}\left(s\right)h_{\text{d}}\left(s\right)+K{g}_{\text{n}}\left(s\right)h_{\text{n}}\left(s\right)}$

where the coefficients of the polynomials $g_{\alpha }\left(s\right)$ and $h_{\alpha }\left(s\right)$ are real.

The characteristic equation is:

$g_{\text{d}}\left(s\right)h_{\text{d}}\left(s\right)+K{g}_{\text{n}}\left(s\right)h_{\text{n}}\left(s\right)=0$ .

which we can write as:

$a_0{s}^n+a_1{s}^{n-1}+a_2{s}^{n-2}+\mathrm{...}+a_{n-1}s+a_n=0$

To apply Routh's criterion, make a table:

The Routh criterion is applied by examining the sign of the coefficient in the column headed by a₀. The number of sign changes in the elements of this column, taken in order, is equal to the number of roots of the characteristic equation that have positive real parts.

Example 1

$s^4+2s^3+3s^2+Ks+2=0$

The table looks like this:

Stability requires that there be no sign changes in the first column. Thus stability requires that $6>K$ and $6K-K^2-8>0$ . These conditions together imply that THE STABILITY REQUIREMENT IS $4>K>2$ .

Consider the critical case $K=4$ . The Routh array is then

The auxilliary equation is defined by the row preceding the zero row, in this case

$s^2+2=0$

Roots of this equation are also roots of the characteristic equation. Thus $s=\pm j\omega =\pm j\sqrt{2}$ are roots of the characteristic equation. The system goes unstable at a frequency of $\omega =\sqrt{2}$ .

Example 2:

$s^4+2s^3+3s^2+6s+2=0$

The only change is the coefficient of s, which is now 6, instead of K. This gives a zero in the leading column of the Routh table. Replace it with a small positive number ε.

The table looks like this:

So in this example also there are two sign changes in the leading column, and therefore two poles with positive real parts. THE SYSTEM IS UNSTABLE.

What's next?

Go on to page 3, where we'll look at steady state errors.