218023 dynamic system and control lecture 2

Unit-Step Response of First-Order Systems We obtain Taking the inverse Laplace transform... Unit-Ramp Response of First-Order Systems We obtain Taking the inverse Laplace transform Th

Linear System Theory

Response Analysis of Systems

• In analyzing and designing control systems, we must have a basis of comparison of performance of various control

systems

• This basis may be set up by specifying particular test input

signals and by comparing the responses of various systems

to these input signals

• Many design criteria are based on the response to such test

signals

• The use of test signals enables one to compare the

performance of many systems on the same basis

Typical Test Signals

• The commonly used test input signals are step functions, ramp

functions, impulse functions, sinusoidal functions, and white noise

• Which of these typical input signals to use for analyzing system

characteristics may be determined by the form of the input that the system will be subjected to most frequently under normal

operation

• The time response of a control system consists of two parts: the

transient response and the steady-state response

• By transient response, we mean that which goes from the

initial state to the final state

• By steady-state response, we mean the manner in which the

system output behaves as time approaches infinity

Step response Harmonic response

• A system is BIBO (bounded-input bounded-output) stable if

every bounded input produces a bounded output

A SISO system is BIBO stable if and only if its impulse response

g(t) is absolutely integrable in the interval [0,∞), i.e.,

� 𝑔𝑔(𝜏𝜏) 𝑑𝑑𝜏𝜏 ∞

for some finite constant M ≥ 0

The input-output relationship is given by the differential equation

𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑑𝑑𝑡𝑡 +

1

𝑇𝑇 𝑑𝑑 𝑡𝑡 = 𝑟𝑟 𝑡𝑡 Ex: RC circuit and thermal system

The transfer function is

The output of the system can be obtained as

Taking inverse Laplace transform gives

Unit-Step Response of First-Order Systems

We obtain

Taking the inverse Laplace transform

Unit-Step Response of First-Order Systems

Block diagram

Error signal

𝑒𝑒 𝑡𝑡 = 𝑟𝑟 𝑡𝑡 − 𝑑𝑑 𝑡𝑡 = 𝑒𝑒 −𝑡𝑡 𝑇𝑇 ⁄

𝑡𝑡 → ∞, 𝑒𝑒(𝑡𝑡) → 0

Unit-Ramp Response of First-Order Systems

We obtain

Taking the inverse Laplace transform

The error signal e(t) is then

e(∞)=T

Unit-Ramp Response of First-Order Systems

We consider a servo system as an example of a second-order system

The system consists of a proportional controller and load elements

(inertia and viscous-friction elements) Suppose that we wish to control

the output position c(t) in accordance with the input position r(t)

Dynamic equation of the servo system

Taking Laplace function yields

The closed-loop transfer function is then obtained as

The closed-loop transfer function possesses two poles is called a order system

second-𝜔𝜔 𝑛𝑛 2 , undamped frequency , attenuation

Damping ratio:

Taking inverse Laplace transform

c(t)

with

Taking inverse Laplace transform

Taking inverse Laplace transform

Approximate form

Definitions of Transient-Response Specifications

Frequently, the performance characteristics of a control system are

specified in terms of the transient response to a unit-step input, since it is

easy to generate and is sufficiently drastic

Delay time

Time required for the response to reach half the final value the very first time

Definitions of Transient-Response Specifications

Definitions of Transient-Response Specifications

Definitions of Transient-Response Specifications

Peak time

The time required for the response to reach the first peak of the overshoot

Definitions of Transient-Response Specifications

Maximum overshoot

The maximum peak value of the response curve measured from unity

Definitions of Transient-Response Specifications

Settling time

Time required for the response curve

to reach and stay within a range about the final value of size specified by

absolute percentage of the final value (usually 2% or 5

Q: What is the best unit-step response?

Second-Order Systems and Transient-Response Specifications

Rise time calculation

Second-Order Systems and Transient-Response Specifications

Peak time calculation

Second-Order Systems and Transient-Response Specifications

Maximum overshoot calculation

Second-Order Systems and Transient-Response Specifications

Settling time calculation

Underdamped system:

Ovedamped system:

Two definitions of ts

Settling time ts versus ζ curves

Mp versus ζ curve

Impulse Response of Second-Order Systems

Taking inverse Laplace transform yields

Impulse Response of Second-Order Systems

Transient Response of Higher-Order Systems

Transfer function

with

poles

zeros

With unit-step input

Closed-loop poles are all real and distinct

Transient Response of Higher-Order Systems

Transfer function

with

poles

zeros

With unit-step input

Real poles and pairs of conjugate poles

complex If any of poles lie in the right-half s plane, a system become unstable

- If all closed-loop poles lie in the left-half s plane, a system is stable

- Whether a linear system is stable or unstable is a property of the

system itself and does not depend on the input or driving function of the system

- The relative stability and transient-response performance of a

closed-loop control system are directly related to the closed-loop pole-zero

configuration in the s plane

Consider the transfer function of the closed-loop system

with

Q: How to determine the stability of the system?

Routh’s Stability Criterion:

- Routh’s stability criterion tells us whether or not there are unstable roots in a polynomial equation without actually

solving for them

- The use of this criterion to collect information about

absolute stability obtained directly from the coefficients of the characteristic equation

• This is for LTI systems with a polynomial denominator

(without sin, cos, exponential etc.)

• It determines if all the roots of a polynomial

- lie in the open LHP (left half-plane),

- or equivalently, have negative real parts

• It also determines the number of roots of a polynomial in

the open RHP (right half-plane)

• It does NOT explicitly compute the roots

Routh’s Stability Criterion

Routh’s Stability Criterion

The number of roots of q(s) with positive real parts is equal to

the number of sign changes in the first column

Examples

If 0 appears in the first column of a nonzero row in Routh

array, replace it with a small positive number

Examples