Control Systems, Poles and Zeros

Poles and Zeros

Poles and Zeros of a transfer function are the values of the complex frequency variable ( s ) for which the transfer function becomes infinite (poles) or zero (zeros), respectively. Specifically, zeros are the values of ( s ) that make the numerator of the transfer function zero, and poles are the values of ( s ) that make the denominator of the transfer function zero. 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 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}^{l t} u ( t ) + B {e}^{m t} u ( t ) + C {e}^{n t} 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}^{l t} u ( t ) = A {e}^{( {σ}_{l} + j {ω}_{l} ) t} u ( t ) = A {e}^{{σ}_{l} t} {e}^{j {ω}_{l} t} u ( t )

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

A {e}^{{σ}_{l} t} [ cos ⁡ ( {ω}_{l} t ) + j sin ⁡ ( {ω}_{l} t ) ] u ( t )

If a complex pole is present it is always accompanied by another pole that is its complex conjugate. The imaginary parts of their time domain representations thus cancel and we are left with 2 of the same real parts. Assuming that the complex conjugate pole of the first term is present, we can take 2 times the real part of this equation and we are left with our final result:

{}2 A {e}^{{σ}_{l} t} cos ⁡ ( {ω}_{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:

• if σl = 0, the response of the pole is a perfect sinusoidal (an oscillator)
• if ωl = 0, the response of the pole is a perfect exponential.
• if σl < 0, the exponential part of the response will decay towards zero.
• if σl > 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.

The polynomial order of a function is the value of the highest exponent in the polynomial.

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 than poles, we can state that the polynomial order of D(s) must be greater than or equal to the polynomial order of N(s).

Example

Consider the transfer function:

H ( s ) = \frac{s + 2}{{s}^{2} + 0.25}

We define N(s) and D(s) to be the numerator and denominator polynomials, as such:

N ( s ) = s + 2

D ( s ) = {s}^{2} + 0.25

We set N(s) to zero, and solve for s:

N ( s ) = s + 2 = 0 → s = − 2

So we have a zero at s → -2. Now, we set D(s) to zero, and solve for s to obtain the poles of the equation:

D ( s ) = {s}^{2} + 0.25 = 0 → s = + i \sqrt{0.25} , − i \sqrt{0.25}

And simplifying this gives us poles at: -i/2 , +i/2. Remember, s is a complex variable, and it can therefore take imaginary and real values.

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.
Addition of poles to the transfer function has the effect of pulling the root locus to the right, making the system less stable. Addition of zeros to the transfer function has the effect of pulling the root locus to the left, making the system more stable.

Second-Order Systems

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

[Second-order transfer function]

H ( s ) = \frac{K {ω}^{2}}{{s}^{2} + 2 ζ ω s + {ω}^{2}}

Where K is the system gain, ζ is called the damping ratio of the function, and ω is called the natural frequency of the system. ζ and ω, if exactly known for a second order system, the time responses can be easily plotted and stability can easily be checked. More information on second order systems can be found here.

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 slower settling time. Damping is the inherent ability of the system to oppose the oscillatory nature of the system's transient response. Larger values of damping coefficient or damping factor produces transient responses with lesser oscillatory nature.

Natural Frequency

The natural frequency is occasionally written with a subscript:

ω → {ω}_{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. Also, ω \ = \ {ω}_{n} when ζ \ = 0.

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