POLES AND ZEROS

In this example we will once again consider the motion of a truck seat equipped with suspension mounted inside a truck which has a suspension of its own.

The most convenient way to analyze a fourth order system is to determine the poles, zeros, and DC Gain of the transfer function, or to determine specific characteristics of the frequency response using the bode plot.  

POLES AND ZEROS

The poles of the system occur when the value of the transfer function approaches infinity, or equivalently when the denominator equals zero.  The poles can be found using MATLAB's pole function.

>> pole(higher_tf)

ans =

-29.7661
-4.0084 + 7.5498i
-4.0084 - 7.5498i
-0.3372

For this system there are two real poles and one pair of complex conjugates. 

The zeros of the system occur when the value of the transfer function equals zero, or equivalently when the numerator equals zero.  The zeros can be found using MATLAB's tzero function.

>> tzero(higher_tf)

ans =

-9.1667
-0.3333

Notice that in this example that one of the poles and one of the zeros are nearly in the same location.  Essentially this pole at -0.337 and this zero at -0.333 cancel each other.  

This pole zero cancellation is very clearly illustrated by plotting both the poles and zeros in the complex plane.  This is easily done using MATLAB's pzmap function.

>> pzmap(higher_tf)

NATURAL FREQUENCY

If a system is second order, the the entire system is characterized by the natural frequency, resonant frequency, and damping ratio of the entire system.  For a higher order system, the terms natural frequency, resonant frequency, and damping ratio apply to each pair of complex conjugate poles.  There may be any number of natural frequencies, depending on how many complex conjugate poles there are.  The natural frequency is equal to the magnitude of each complex conjugate pole. 

For this system there is one pair of complex conjugates, thus there is only one natural frequency.  The magnitude of each complex conjugate pair is found by using the Pythagorian theorem.  

>> wn = sqrt(4.0084^2 + 7.5498^2)

wn =

8.5479

RESONANT FREQUENCY

The resonant frequency is is related to the natural frequency, but not exactly the same.  Unlike the natural frequency, the resonant frequency is affected by the damping in the system.  The resonant frequency in rad/s is equal to the size of the imaginary part of the complex poles.

For this system there is only one pair of complex conjugate pairs, so there is one resonant frequency at 7.5498 rad/s.

DAMPING RATIO 

The damping ratio can be found using MATLAB's damp function.

>> damp(higher_tf)

Eigenvalue               Damping Freq.  (rad/s)

-3.37e-001               1.00e+000      3.37e-001
-4.01e+000 + 7.55e+000i  4.69e-001      8.55e+000
-4.01e+000 - 7.55e+000i  4.69e-001      8.55e+000
-2.98e+001               1.00e+000      2.98e+001 

The damping ratio for the complex eigenvalue can also be manually calculated as the ratio of the real part of the pole to the natural frequency.

>> damp = 4.0084 / sqrt(4.0084^2 + 7.5498^2)

damp =

0.4689

DC GAIN

Logically the DC gain of a suspended truck seat must equal 1 since the truck and seat always returns to the same position relative to ground after a bump.  This is a good check for a model of this type.

MATLAB can find the DC gain of a system using the dcgain function. 

>> dcgain(higher_tf)

ans =

1

There are two other ways to determine the DC gain of this system.  The first is to look at the bode plot and visually inspect the magnitude of the plot when the frequency approaches zero.  

DC Gain = 10^M(0)/20

The magnitude as frequency approaches zero is also zero, thus:

>> dcgain = 10^(0/20)

dcgain =

1

The other method is to determine the value of the transfer function when s equals zero:

Assigning s=0 and plugging into the expression for the transfer function yields:

>> s=0;
>> (b2*s+k2)*(b1*s+k1)/((m1*s^2 + (b1+b2)*s + (k1+k2))*(m2*s^2+b2*s+k2)-(b2*s+k2)^2)

ans =

1

SETTLING TIME
Strictly speaking, there is no closed form expression for the settling time because the order of the system is greater than two.  However the settling time can be determined by observing when the step response stays within 2% of the final value. 
With a final value of 1, the 2% settling time occurs when a peak or dip if oscillation falls above 0.98 and below 1.02.

From the step response, the 2% settling time appears to be around 1.15 seconds.

Carnegie Mellon University | University of Michigan

Mechanical Engineering Department