Control Response Patterns

Control Response Patterns

Engr. R. L. Nkumbwa

May, 2009

04/20/23 10:15 PMEng. R. L. Nkumbwa @CBU

20102

System Metrics and Time-Domain Analysis

System Metrics Time-Domain Analysis

– Time Response– Poles and Zeros– Transient Response


20103

System Metrics

So, What are System Metrics? When a system is being designed and

analyzed, it doesn't make any sense to test the system with all manner of strange input functions or to measure all sorts of arbitrary performance metrics.

Instead, it is in everybody's best interest to test the system with a set of standard, simple, reference functions.


20104

Control Systems


20105


20106

System Metrics

Once the system is tested with the reference functions, there are a number of different metrics that we can use to determine the system performance.

So, what are the examples of such metrics?


20107

Time-Response Analysis

Since time is used as an independent variable in most control systems, it is usually of interest to evaluate the state and the output responses with respect to time or simply, the Time-Response.

In control system design analysis, a reference input signal is applied to a system and the performance of the system is evaluated by studying the system response in the time-domain.


20108

Time-Response

The time-response of a control system is usually divided into two parts namely; the Steady-State Response and the Transient Response.

In other words, the output response of a system is the sum of two responses: the forced response (steady-state response) and the natural response (zero-input response).


20109

Time-Response of an Elevator


201010

Transient Response

Defined as the part of the time response that goes to zero as time goes to infinity.


201011

Steady-State Response

Defined as the part of the total response that remains after the transient has died out.


201012

Poles and Zeros

The poles of a transfer function are:– (1) The values of the Laplace transform variable, s ,

that cause the transfer function to become infinite, or – (2) Any roots of the denominator of the transfer

function that are common to roots of the numerator. The zeros of a transfer function are:

– (1) The values of the Laplace transform variable, s , that cause the transfer function to become zero, or

– (2) Any roots of the numerator of the transfer function that are common to roots of the denominator.


201013


201014

Response blueprint

1. A pole of the input function generates the form of the forced response.

2. A pole of the transfer function generates the form of the natural response.

3. A pole on the real axis generates an exponential response.

4. The zeros and poles generate the amplitudes for both the forced and natural responses.


201015

)5)(4)(2(

3

sss

sssR

1)( )(sC

542)( 4321

s

k

s

k

s

k

s

ksC

Natural responseForced response

ttt ekekekktc 54

43

221)(


201016

Standard Input Signals

All of the standard inputs are zero before time zero. All the standard inputs are causal.

So, what is causal? Causal: A system whose output does not

depend on future inputs. All physical systems must be causal.


201017

Standard Input Signals

There are a number of standard inputs that are considered simple enough and universal enough that they are considered when designing a control system.

These inputs are known as a unit step, a ramp, and a parabolic input functions.


201018

Unit Step Function

A unit step function is defined piecewise as such:

The unit step function is a highly important function, not only in control systems engineering, but also in signal processing, systems analysis, and all branches of engineering.


201019

Unit Step Function


201020

Ramp Input Function

A unit ramp is defined in terms of the unit step function, as such: r(t) = tu(t).

It is important to note that the ramp function is simply the integral of the unit step function:


201021

Ramp Input Function


201022

Parabolic Input Function

A unit parabolic input is similar to a ramp input:

Notice also that, the unit parabolic input is equal to the integral of the ramp function:


201023

Parabolic Input Function


201024

Steady State

To be more precise, we should have taken the limit as t approaches infinity. However, as a shorthand notation, we will typically say "t equals infinity", and assume the reader understands the shortcut that is being used.


201025

Steady State

When a unit-step function is input to a system, the steady state value of that system is the output value at time t = ∞.

Since it is impractical (if not completely impossible) to wait till infinity to observe the system, approximations and mathematical calculations are used to determine the steady-state value of the system.


201026

Steady State

Most system responses are asymptotic, that is, the response approaches a particular value. Systems that are asymptotic are typically obvious from viewing the graph of that response.


201027

Step Response

The step response of a system is most frequently used to analyze systems and there is a large amount of terminology involved with step responses.

When exposed to the step input, the system will initially have an undesirable output period known as the transient response.

The transient response occurs because a system is approaching its final output value.

The steady-state response of the system is the response after the transient response has ended.


201028

Step Response

It is common for a systems engineer to try and improve the step response of a system.

In general, it is desired for the transient response to be reduced.


201029

First-Order Systems


201030

Initial Conditions are zero


201031

First-Order Systems Response


201032

System Response

K (1 − e−t /τ )

System response. K = gain

Response to initial condition


201033

Unit Step Response

atnf etctctc 1)()()(

)()()()(

ass

asGsRsC


201034

The time constant can be described as the time for to decay to 37% of its initial value. Alternately, the time is the time it takes for the step response to rise to 67% of its final value.

The reciprocal of the time constant has the units (1/seconds), or frequency. Thus, we call the parameter a the exponential frequency.

Unit Step Response


201035

Time Constant

37.01

1

eeat

at

63.037.011)(1

1

at

at

atetx


201036

Second-Order Systems Response

ζ = 0


201037


201038

Step Response Vs. Pole Location


201039

System Response


201040

Target Value

The target output value is the value that our system attempts to obtain for a given input.

This is not the same as the steady-state value, which is the actual value that the target does obtain.

The target value is frequently referred to as the reference value, or the "reference function" of the system.

In essence, this is the value that we want the system to produce.


201041

Example of an Elevator

When we input a "5" into an elevator, we want the output (the final position of the elevator) to be the fifth floor.

Pressing the "5" button is the reference input, and is the expected value that we want to obtain.

If we press the "5" button, and the elevator goes to the third floor, then our elevator is poorly designed.


201042

Time-Domain Specifications

So, what are Time-Domain Specifications?


201043

Time-Domain Specifications


201044

Rise Time

Is the amount of time that it takes for the system response to reach the target value from an initial state of zero.

Rise time is defined as the time for the waveform to go from 0.1 to 0.9 of its final value.

Rise time is typically denoted tr, or trise. This is because some systems never rise to

100% of the expected, target value and therefore, they would have an infinite rise-time.


201045

Settling Time

After the initial rise time of the system, some systems will oscillate and vibrate for an amount of time before the system output settles on the final value.

The amount of time it takes to reach steady state after the initial rise time is known as the settling time

Which is defined as the time for the response to reach and stay within, 2% (or 5%) of its final value.

Damped oscillating systems may never settle completely.


201046

Settling time

nnst

4102.0ln( 2


201047

Peak Time

The time required to reach the first or maximum peak. 21

n

pt

22

2

2)()]([

nn

n

ssssCtcL

)1()(

11

222

2

2

nn

nn

s


201048

Percent Overshoot

The amount that the waveform overshoots the steady-state or final value at the peak time, expressed as a percentage of the steady-state value.

100% max

final

final

c

ccOS


201049

Pole-Zero Plots

)1)(1(2)(

22

2

22

2

nnnn

n

nn

n

jsjssssG


201050

Step Response as Poles Moves


201051

Example of a Refrigerator

The refrigerator has cycles where it is on and when it is off. When the refrigerator is on, the coolant pump is running, and the temperature inside the refrigerator decreases.

The temperature decreases to a much lower level than is required, and then the pump turns off. When the pump is off, the temperature slowly increases again as heat is absorbed into the refrigerator.

When the temperature gets high enough, the pump turns back on. Because the pump cools down the refrigerator more then it needs to initially, we can say that it "overshoots" the target value by a certain specified amount.


201052

Exercise

Find peak time, percent overshoot, and settling time from pole location.


201053

Exercise


201054

Solution

2173 nndd jjj

dn

pt

21

100% )1/( 2

eOS

dnst

44


201055

Effects of adding a Zero


201056


201057

Time-Domain Analysis

Do further research on this topic please. Consider researching on Positional error

constants

Control Response Patterns

Documents

Transcript of Control Response Patterns