U NDERDAMPED R ESPONSES

Second-order underdamped responses exhibit an over and undershoot signal (oscillating response) at a natural resonant frequency of ω n in radians/ second. This oscillation is the result of a damping factor ( ζ ) that is less than

1. This means that instead of being able to factor the denominator of the second-order lag transfer function into a polynomial (i.e., s 2 +2 ζ ω n s+ ω n 2 ), the denominator becomes a complex-root quadratic equation. The inverse Laplace transform of this equation produces an exponential, decreasing

sinusoidal response to a unit step input ( 1 s ) represented by (from Table 14-2):

  For small values of ζ , this response exhibits a behavior to a unit step

 s  s + 2 ζω n s + ω n  

approximate to:

− ζω n Out t

() t = A 1 + e sin( ω n t )

unit step

This makes the transfer function response approximate to:

− ζω n H t

() t ≈ e sin( ω n t ) ≈ sin ( ω n t ) ζ = 0

which is the form of a sine curve. The response of this equation, shown in Figure 14-55, illustrates the damping factor ζ of the sinusoidal response. The

closer the damping factor is to 1 (critical damping), the lower the frequency of oscillation and the sooner it will level off (see Figure 14-56a). Remember that if ζ = 0, the response will oscillate forever as a sinusoidal response at a

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– ζω e t n (Damping)

t Figure 14-55. Underdamped response.

frequency ω n ; therefore, the closer the value of ζ gets to zero (see Figure 14- 56b), the higher the frequency and the longer the oscillations will last ( τ becomes longer).

– e ζω n t

( ζ is closer to 1)

(a)

– ζω n e t

( ζ is closer to 0)

(b)

Figure 14-56. Frequency of oscillation is (a) lower when ζ is closer to 1 and (b) higher

when ζ is closer to 0.

The exponential sinusoidal response of an underdamped second-order system will settle to 5% of its steady-state value within 3 τ (three time

constants), to 2% within 4 τ , and to 0.5% within 5 τ . The second-order lag response ( τ sys ) for an underdamped system is defined as:

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sys =

ζω n

which indicates that the lag time constant depends on the value of ζ . Figure 14-57 illustrates some typical parameters used to describe underdamped second-order systems.

Underdamped Second-Order Process

H 1.05 A Overshoot

Limits

( s)

A 0.95 A A 0.9 A

Step with amplitude A

A = Amplitude after all gains;

value is at steady state

A p = Peak value of the overshoot t 0.5 A

= Time required for the response

to be within tolerance values (e.g., 5%)

t p = Time to peak value t r = Rise time—the time to get

from 10% to 90% of the final steady-state value

0.1 A t = Time constant t—the time the

system takes to reach the

value of 1/ e of A (steady-state)

t d = Time delay—the time interval

required to reach half of the

steady-state value (0.5

A) after

an application of input or

disturbance change

Figure 14-57. Parameters of an underdamped second-order system.

E X AM PLE 1 4 -7

Compare the relationship between the first-order time response term − t

e −ζω n and the second-order, sinusoidal, exponential decay term t e , which is used in the underdamped transfer function:

e t − ζω n sin( ω

S OLU T I ON

The time constant τ sys for an underdamped system is equal to:

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The response of a first-order exponential system is e τ , where τ is the −ζω n t system’s time constant. Therefore, the decaying term e in the

equation:

− e ζω n t sin( ω

is equal to:

− τ t sys

e sin( ω n t )

This indicates that, in an underdamped second-order system, the value of τ sys (lag time) becomes larger as the value of ζ becomes

smaller ( τ

1 sys = ζω n ) This is similar to the behavior of a first-order system with a long lag, because the time to reach the steady-state value will be long. In a second-order underdamped system, the oscillation − t τ continues for a longer time since the term sys e provides less damping.

If 1 ζ = 1, the value of τ

sys becomes ω n , which is the lag time of a critically

damped system.

1 4 -8 S U M M A RY

The objective of a process control system is to maintain the process variable (process output) at a desired target value, referred to as the set point. The system provides this control by implementing a feedback loop, meaning that it reads the process variable and compares it to the set point value. The controller then uses the difference between these values, as computed by E = SP – PV , to determine how much corrective action it must take. The error, which the controller calculates as a percentage of the full range of the process variable, can be caused by changes in the set point or by disturbances to the process.

Open-loop systems are systems in which the process variable is not fed back into the control system for reference. Closed-loop systems, on the other hand, do receive process variable feedback. Most process control systems are closed-loop systems that receive negative feedback. In a negative feedback system, the controller determines the error by subtracting the process variable from the set point.

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Process dynamics refers to changes in the process that occur due to disturbances or changes in the set point. Process gain changes are a result of gains in the process variable value created by changes in the control variable output. The dynamics of a process also includes dead time and lag time. Dead time is the delay that occurs between the moment a change is made in the control variable and the moment the process variable begins to react to the control variable change. Lag time is the delay associated with the time required by the process control loop to bring the process variable to the set point by adjusting the final control element. The lag time is a finite time required by the control system to physically adjust the final control element (e.g., a steam valve).

A transient is the process variable response to a change in set point or to the creation of a disturbance (e.g., a load change). The transient response depends not only on the dynamics of the process, but also on the characteristics of the process itself. These characteristics are the result of the transfer functions of the controller and the process. A transfer function is the mathematical representation of a system’s response, where the response is computed by dividing the output by the input. Transfer functions are expressed in the frequency domain using Laplace transforms, to allow easy algebraic manipu- lations of the equations. The inverse Laplace transform of a transfer function converts a frequency-based Laplace response into a time-based response.

Each element in a control system loop has a transfer function associated with it—the controller has one and the process has one. The combined controller/ process system also has a transfer function. Transfer functions are categorized as either first-order or second-order responses. First-order systems have one lag time associated with the process, while second-order systems have two lag times. Laplace transforms are used to mathematically represent both first- and second-order process transfer functions, as well as controller transfer functions and the combination of both process and controller functions in a closed-loop configuration. Although it is difficult to obtain the actual transfer function of a process, a knowledge of the type of transfer function expected from a process response is extremely useful, especially when tuning the controller.

First-order systems have one lag time, resulting in an exponential, decaying response. When the system receives a step input change, its open-loop output will have the following time domain response, which smoothly follows the input:

out = V in 1 − e () τ

The time constant τ specifies the time the output takes to achieve 63.2% of the final steady-state value. The time constant τ is sometimes referred to as the 63% response time. After 5 τ periods have elapsed, the value of the output

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response will be at 99.33% of its final value. In Laplace form, the transfer function of a first-order system has the form:

Second-order systems have two lag times and are described by the transfer equation:

Out 2 A ω

H n ()

( s + 2 ζω n s + ω n )

Second-order systems can have three types of responses, depending on their damping coefficient: overdamped ( ζ > 1), critically damped ( ζ = 1), and underdamped ( ζ < 1). Each of these types of responses have different inverse Laplace transforms, translating into different time domain responses.

Overdamped responses ( ζ > 1) have two different time constants ( τ 1 and τ 2 ) , and their response over time in reaching the set point is sluggish. Critically damped systems ( ζ = 1) have two lag times, or time constants, that are equal

( τ 1 = τ 2 ) . These systems reach the set point much faster than overdamped systems. Underdamped responses ( ζ < 1) produce a faster response than either overdamped or critically damped responses, resulting in an overshoot and undershoot of the final value that dies off exponentially as the steady-state value is approached. An underdamped response has two time constants that are imaginary, or mathematically speaking, that have complex roots pro- duced by their quadratic equation.

Although some processes have more complicated responses (third- and fourth-order responses), these processes’ transfer functions can be approxi- mated by second-order system transfer functions. Most manufacturing processes can be classified as either first-order or second-order systems. In the next chapter, we will discuss how to use PID control to adjust the inputs to these complex systems to obtain a desired output.

K EY control element T ERMS control loop

control variable critically damped response dead time error error deadband first-order response lag time Laplace transforms

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overdamped response process control process gain process variable second-order response set point steady state step response step test transfer function transient response underdamped response

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