Lecture 4: Second-Order Open-Loop Systems

A __________________ is one whose output, y(t), is described by the solution of a second-order differential equation.

second-order system

Parameters in Second Order Systems

τ = Natural period of oscillation of the system

ζ = damping factor

K_p= steady gain

Systems with second- or higher-order dynamics can arise from several physical situations, which can be classified into three categories:

• Multi-capacity Processes

• Inherently 2nd-order Systems

• Processing Systems w/ Controller

processes that consist of two or more capacities (first-order systems) in series, through which material or energy must flow,

Multi-capacity processes

such as the fluid or mechanical solid components of a process that possess inertia and are sub- jected to acceleration.

Inherently 2nd order systems

Such systems are rare in chemical processes.

Inherently 2nd order systems

May exhibit second- or higher-order dynamics. In such cases, the controller which has been installed on a processing unit introduces additional dynamics which, when coupled with the dynamics of the unit, give rise to second- or higher-order behavior.

Processing Systems w/ Controller

The very large majority of the second- or higher-order systems encountered in a chemical plant come from _______________

multicapacity processes (or effect of process control systems)

form of the response y(t) will depend on the location of the ___________________ in the complex plane

two poles, p₁ and p₂

Three Cases of Second-Order Systems based on the Damping Factor (ζ) Values

CASE A: When ζ > 1, two distinct and real poles

CASE B: When ζ = 1, two equal poles (multiple pole)

CASE C: When ζ < 1, two complex conjugate poles

Case A is also known as

Overdamped response

resembles a little the response of a first-order system to a unit step input.

Overdamped response

But when compared to a first-order response we notice that the system initially delays to respond and then its response is rather sluggish.

Overdamped response

Overdamped response becomes slower and sluggish as ___ increases

ζ

Finally, we notice that as time goes on, the response approaches its ultimate value asymptotically.

Overdamped response

Overdamped are the responses of ______________________

multi-capacity processes

CASE B is also called

Critically damped response

a second- order system with _______________ approaches its ultimate value faster than does an overdamped system.

critical damping

CASE C is also called

Underdamped response

The ______________________ is initially faster than the critically damped or overdamped responses

underdamped response

Although the underdamped response is initially faster and reaches its ultimate value quickly, it does not stay there, but it ________________________________

starts oscillating with progressively decreasing amplitude.

The oscillatory behavior becomes more pronounced with smaller values of the ___________________—

damping factor, ζ

It must be emphasized that almost all the underdamped responses in a chemical plant are caused by the interaction of the _______________with the process units they control.

controllers

It is the ratio A/B, where B is the ultimate value of the response and A is the maximum amount by which the response exceeds its ultimate value.

Overshoot

It is the ratio C/A (i.e., the ratio of the amounts above the ultimate value of two successive peaks).

Decay Ratio

radian frequency (rad/time) of the oscillations

Period of oscillation

A second-order system with ζ=0 is a system free of any damping.

Natural period of oscillating

it will oscillate continuously with a constant amplitude and a natural frequency

Natural period of oscillating

The response of an underdamped system will reach its ultimate value in an oscillatory manner as t approaches infinity

Response time

For practical purposes, it has been agreed to consider that the response reached its final value when it came within ____ of its final value and stayed there.

+-5%

The time needed for the response to reach the response’s final value is known as the

response time

Different characteristics of underdamped response

This term is used to characterize the speed with which an underdamped system responds.

Rise time

It is defined as the time required for the response to reach its final value for the first time

Rise time

Relationship between ζ and rise time

the smaller the value of ζ, the shorter the rise time, but at the same time the larger the value of the overshoot.

___________________ do not have to involve more than one physical processing unit.

Multicapacity processes

the first system affects the second by its output, but it is not affected by it

Noninteracting Capacities

|[f the time constants Tp1 and Ty are equal, the two poles are equal, therefore, noninteracting capacities always result in an ___________________ second order system.

overdamped or critically damped

The response of the overdamped multicapacity system to step input change is ___________

S-shaped

This is in contrast to a ____________, which has the largest rate of change at the beginning.

first-order response

Sluggishness in overdamped systems is also called

Transfer lag

This sluggishness or delay is also known as transfer lag and is characteristic of ___________________

multicapacity systems.

Relationship between number of capacities and sluggishness

As the number of capacities in series increases, the delay in the initial response (sluggishness) becomes more pronounced.

This is the distinguishing characteristic of ____________________ and indicates the mutual effect of the two capacities.

interacting capacities

it is noticeable that they differ only in the coefficient of s in the denominator by the term A1R2 (a.k.a. ___________________)

interaction factor

Relationship between interaction factor and interaction between tanks

the larger the value of A1R2 the larger the interaction between the two tanks.

Which is more sluggish, interacting or noninteracting systems?

Interacting capacities are more sluggish than the noninteracting.

Such a process can exhibit underdamped behavior. and consequently it cannot be decomposed into two first-order systems in series

Inherently Second-Order Processes

Inherently 2nd order systems occur rather rarely in a chemical process, and they are associated with the motion of liquid masses or the mechanical translation of solid parts, possessing:

• (1) inertia to motion

• (2) resistance to motion

• (3) capacitance to store mechanical energy.

The value of the manipulated variable is determined by two terms, one of which is proportional to the error h', and the other proportional to the time integral of the error.

proportional-integral control

The control action described by eq. {11.36) is called proportional-integral control, because the value of the manipulated variable is determined by two terms:

• proportional to the error h

• proportional to the time integral of the error

Systems with inherent second-order dynamics can exhibit _________________________

oscillatory (underdamped) behavior