Control Systems/Poles and Zeros

Poles and Zeros

Poles and Zeros of a transfer function are the frequencies for which the value of the transfer function becomes infinity or zero respectively. The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Control systems, in the most simple sense, can be designed simply by assigning specific values to the poles and zeros of the system.

Physically realizable control systems must have a number of poles greater than or equal to the number of zeros. Systems that satisfy this relationship are called proper. We will elaborate on this below.

Time-Domain Relationships

Let's say that we have a transfer function with 3 poles:

$H(s)=\frac{a}{(s+l)(s+m)(s+n)}$

The poles are located at s = -l, -m, -n. Now, we can use partial fraction expansion to separate out the transfer function:

$H(s)=\frac{a}{(s+l)(s+m)(s+n)}=\frac{A}{s+l}+\frac{B}{s+m}+\frac{C}{s+n}$

Using the inverse transform on each of these component fractions (looking up the transforms in our table), we get the following:

$h(t)=A{e}^{-lt}u(t)+B{e}^{-mt}u(t)+C{e}^{-nt}u(t)$

But, since s is a complex variable, l m and n can all potentially be complex numbers, with a real part (σ) and an imaginary part (jω). If we just look at the first term:

$A{e}^{-lt}u(t)=A{e}^{-({\sigma }_l+j{\omega }_l)t}u(t)=A{e}^{-{\sigma }_{l}t}{e}^{-j{\omega }_{l}t}u(t)$

Using Euler's Equation on the imaginary exponent, we get:

$A{e}^{-{\sigma }_{l}t}[cos({\omega }_{l}t)-j\sin ({\omega }_{l}t)]u(t)$

And taking the real part of this equation, we are left with our final result:

$A{e}^{-{\sigma }_{l}t}cos({\omega }_{l}t)u(t)$

We can see from this equation that every pole will have an exponential part, and a sinusoidal part to its response. We can also go about constructing some rules:

1. if σ_l = 0, the response of the pole is a perfect sinusoid (an oscillator)
2. if ω_l = 0, the response of the pole is a perfect exponential.
3. if σ_l > 0, the exponential part of the response will decay towards zero.
4. if σ_l < 0, the exponential part of the response will rise towards infinity.

From the last two rules, we can see that all poles of the system must have negative real parts, and therefore they must all have the form (s + l) for the system to be stable. We will discuss stability in later chapters.

What are Poles and Zeros

Let's say we have a transfer function defined as a ratio of two polynomials:

$H(s)=\frac{N(s)}{D(s)}$

Where N(s) and D(s) are simple polynomials. Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting N(s) = 0 and solving for s.

Poles are the roots of D(s) (the denominator of the transfer function), obtained by setting D(s) = 0 and solving for s. Because of our restriction above, that a transfer function must not have more zeros then poles, we can state that the polynomial order of D(s) must be greater then or equal to the polynomial order of N(s).

Effects of Poles and Zeros

As s approaches a zero, the numerator of the transfer function (and therefore the transfer function itself) approaches the value 0. When s approaches a pole, the denominator of the transfer function approaches zero, and the value of the transfer function approaches infinity. An output value of infinity should raise an alarm bell for people who are familiar with BIBO stability. We will discuss this later.

As we have seen above, the locations of the poles, and the values of the real and imaginary parts of the pole determine the response of the system. Real parts correspond to exponentials, and imaginary parts correspond to sinusoidal values.

Second-Order Systems

The canonical form for a second order system is as follows:

$H(s)=\frac{{\omega }^2}{s^2+2\zeta \omega s+{\omega }^2}$

Where ζ is called the damping ratio of the function, and ω is called the natural frequency of the system.

Damping Ratio

The damping ratio of a second-order system, denoted with the greek letter zeta (ζ), is a real number that defines the damping properties of the system. More damping has the effect of less percent overshoot, and faster settling time.

Natural Frequency

The natural frequency is occasionally written with a subscript:

$\omega \to {\omega }_n$

We will omit the subscript when it is clear that we are talking about the natural frequency, but we will include the subscript when we are using other values for the variable ω.