Process Control Instrumentation – Basics.

Dynamics of first and second order systemsSystems with inverse responseStability

What is a system?

A system is a combination of components that act together and perform a certain objective

Any item in any systems Can be represented by a block As shown

Identification of the order of the system.

A system may be described by one of the following A transfer function, A linear differential equation with constant

coefficients that relates the input and the output of the system,

An impulse response, A set of state equations,

If you are given a system and an input, you should be able to find the output of the system.

Transfer functions

Transfer functions are generally expressed as a ratio of polynomials

Where

The polynomial is called the characteristic polynomial of

Roots of are the zeroes of Roots of are the poles of

The transfer functions are commonly used in the analysis of single-input single-output filters, for instance. It is mainly used in signal processing, communication theory, and control theory. The term is often used exclusively to refer to linear, time-invariant systems (LTI)

It's often possible to measure the time response of a system to a controlled input.  In a situation like that you can use that measured data to calculate a transfer function for a system. 

The normally used controlled inputs are

1) Step

2) Ramp.

A First Order System

Many measuring elements or systems can be represented by a first order differential equation in which the highest derivatives is of the first order, i.e. dx/dt, dy/dx, etc. For example

where a and b are constants; f(t) is the input; q(t) is the output

An example of first order measurement systems is a mercury-in-glass thermometer.

ttTdt

td

ttdt

td

oio

oio

1

where i and o is the input and output of the thermometer. Therefore, the differential equation of the thermometer is:

tdt

tdTt i

oo

Consider this thermometer is suddenly dipped into a beaker of boiling water, the actual thermometer response (o) approaches the step value (i) exponentially according to the solution of the differential equation

o = i (1- e-t/T)

i

0(t)0(T)~0.632i

Response of a mercury in glass thermometer to a step change in temperature

First order response

The time constant is a measure of the speed of response of the instrument or system

After three time constants the response has reached 95% of the step change and after five time constants 99% of the step change.

Hence the first order system can be said to respond to the full step change after approximately five time constants.

First order process

Ramp response:

Ramp input of slope a

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

First Order Systems

Sinusoidal response

Sinusoidal input Asin(t)

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

2

AR

Second order systems

Very many instruments, particularly all those with a moving element controlled by a spring, and probably fitted with some damping device, are of ‘second order’ type. Systems in this class can be represented by a second order differential equation where the highest derivative is of the form d2x/dt2, d2y/dx2, etc. For example,

ttdt

td

dt

tdion

on

o 2

2

2

2

where and n are constants.

For a damped spring-mass system,

m

kn

(in rad/s)

m

kfn 2

1

(in Hz)

Natural frequency

Response of a second order system to a step input

The parameters you find in a first order system determine aspects of various kinds of responses. Whether we are talking about impulse response, step response or response to other inputs, we will still have the following quantities and system parameters.

x(t)  =  Response of the System, 

u(t)  =  Input to the System, 

t  =  The System Time Constant, 

Gdc  =  The DC Gain of the System.

Every system will have an input which we can call u(t), and a response we will denote by x(t).  Each system will also have a time constand and a DC gain.t, the time constant, will determine how quickly the system moves toward steady state.Gdc, the DC gain of the system, will determine the size of steady state response when the input settles out to a constant value.

      There are some important points to note about the step response of a first order linear system.

When the step is applied, the derivative of the output changes immediately. The size of the derivative change depends upon the size of the

step, but as long as the step is non-zero, the derivative will have a jump.

To get the steady state value, multiply the input step size by the DC Gain. If the input is not a step but if it does reach a steady state value,

the output will be the DC Gain multiplied by the steady state value of the input.

        That's pretty much it for the step response of a first order system.  Now that you know what it looks like it's time to start looking at how you can use this concept!! 

Some Observations on First Order Systems  

Observations on second order systems What you see there is an example of a second order

step response.  It has characteristics - like decaying oscillations - that you can have in second order systems.  Those characteristic decaying oscillations are not to be seen in first order systems.  If you see decaying oscillations, you know you don't have a first order system.  On the other hand, not every second order system will exhibit those decaying oscillations.  Second order systems are more complex than that.

algo

Animation .

We are given the following response curve.

Now, let's look closely at the response we have.  We will focus on features of the response that are informative.

The overall shape of the response appears to be a decaying exponential with an added constant value.  Generally, the shape seems to indicate a single decaying exponential which would indicate one time constant - a first order system.

At t = 0, the slope of the response seems to jump immediately to a positive value.  That's an indication that the system could be first order.

Control System Stability

The stability of a control system is often extremely important and is generally a safety issue in the engineering of a system.  An example to illustrate the importance of stability is the control of a nuclear reactor.  An instability of this system could result in an unimaginable catastrophe!

Definitions

The stability of a system relates to its response to inputs or disturbances. A system which remains in a constant state unless affected by an external action and which returns to a constant state when the external action is removed can be considered to be stable.

A systems stability can be defined in terms of its response to external impulse inputs..

Definition .a : A system is stable if its impulse response approaches zero as time approaches infinity.. 

The system stability can also be defined in terms of bounded (limited) inputs..

Definition .b:A system is stable if every bounded input produces a bounded output.

Stability Notes Control analysis is concerned not only with the stability of a system but also the

degree of stability of a system.. A typical system equation without considering the concept of integral action is of the form.

[ a 2 D 2 + a 1 D + a 0 ].x = f(D) y This is defined as be the highest order of D on the LHS as a equation of order 2.

The transient response, and as a result the stability, of such a system depends on the coefficients a 0, a 1 , a 2. Assuming a 0 >0 then provided that a 1 >0 and a 2 >0 the complementary function will not contain any positive time exponentials and the system will be stable.   If either a 1 < 0 (negative damping) or a 2 < 0 (negative mass) the transient response will contain positive exponentials and the system will be unstable.. 

If a 1 = 0 (As resulting from zero damping) then the complementary function will oscillate indefinitely. This is not an unstable response but this marginally stable response is not satisfactory. Following are a number of plots to illustrate the types of stability responses resulting from an input...

 For the stability analysis it is sufficient to check the poles of the transfer function   of the system, that is the roots   of its characteristic equation

Now the following necessary and sufficient stability conditions can be formulated:

a) Asymptotic stabilityA linear system is only asymptotically stable, if for the

roots   of its characteristic equation     for all     

is valid, or in other words, if all poles of its transfer function lie in the left-half   plane.

b) Instability A linear system is only unstable, if at least one pole of its transfer function

lies in the right-half   plane, or, if at least one multiple pole (multiplicity  ) is on the imaginary axis of the   plane.

c) Critical stability A linear system is critically stable, if at least one single pole exists on the

imaginary axis, no pole of the transfer function lies in the right-half   plane, and in addition no multiple poles lie on the imaginary axis.

Systems with inverse response.

When the initial response of a dynamic system is in a direction opposite to the final outcome, it is called an inverse response.

rises

falls

Why inverse response?

Inverse response typically results when two separate effects are occurring at the same time, but with different directions and dynamics. The system is thus two transfer functions coupled in parallel, so the individual outputs can be added to obtain the overall response.

To better understand inverse response, consider two first order transfer functions connected in parallel (as shown in the block diagram). We will look at the case where Process 1 has smaller time constant and gain than

Process 2.

Because Process 1 is faster, it dominates the initial response, but since Process 2 has a larger magnitude (because of its larger steady state gain), it dominates the steady state response.