Digital Control Systems -...

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Digital Control Systems

Transcript of Digital Control Systems -...

Page 1: Digital Control Systems - faculty.mercer.edufaculty.mercer.edu/jenkins_he/documents/DigitalControlSystems...Digital (Discrete) Controllers Set-up for a typical digital controlled system

Digital Control

Systems

Page 2: Digital Control Systems - faculty.mercer.edufaculty.mercer.edu/jenkins_he/documents/DigitalControlSystems...Digital (Discrete) Controllers Set-up for a typical digital controlled system

Discretization & Digitization of an analog (continuous) signal

• Time sample

– Usually uniform rate

– new value each ΔT

– Discrete samples

– A sequence of numbers

• Level of magnitude limited (Digital Resolution)

– 8-bit =28=256 levels

– 0-5 volts at 8-bit resolution 5V/256= 0.0195313 V= real value of a bit

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Digital (Discrete) Controllers

Set-up for a typical digital controlled system

Sampled values of continuous signals (in/out)

Computer/Digital Controller

Physical system with controller, sensors, actuators

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Continuous and Discrete Control

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Sample and Hold Control, u(t)=(kT) T= sampling time

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Continuous Controllers

Differential equations

All analog signals and devices

Amplifiers, resisters , capacitors, inductors hardwired

Knobs connected to potentiometers for input and interfaces

Not very easy to manipulate and display

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Digital/Discrete Systems

Difference equations

Use digital (computer) devices for control and signal processing

Very flexible to change control

Easy to manipulate and present Inputs/Outputs via computer display

Still need to analog circuits for power to actuator and some output sensor

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Pros and Cons of Digital Signal Processing

• Pros

– Accuracy can be controlled by choosing word length, number of bits

– Many systems inherently digital

– Repeatable

– Control algorithms easily modified

– Sensitivity to electrical noise is minimal

– Flexibility can be achieved with software implementations

– Non-linear and time-varying operations are easier to implement

– Digital storage is cheap

– Price/performance and reduced time-to-market

• Cons

– Sampling causes loss of information

– A/D and D/A requires mixed-signal hardware

– Limited speed of processors

– Quantization and round-off errors

8

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Sampling

Result of sampling a continuous time signal

The result of sampling is a series of values in time.

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Sampled

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Digital Control Systems:

Zero-Order Hold (Sample and hold)

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Sampling time (ΔT)

ΔT has limits on how small it can be

Based on CPU and ALL algorithms

Nyquist Frequency: Limit on highest system frequency (smallest period) that can be detected which is half the sampling frequency (Hz) (2/ ΔT)

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Aliasing effect when using low sampling rate

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The Basic Control System

March 30, 2002 John Y. Hung

Control

Algorithm Plant

setpoint error controller

output

process

variable

+ –

Implemented on a microcontroller …

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The Basic Tasks

• Measure the process variable

• Compute control algorithm

– data scaling

– filtering

– decision making

• Generate controller output signal

REPEAT

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Timing

• Basic Tasks performed periodically

• Sampling frequency rule-of-thumb:

5X-10X closed-loop bandwidth (Hz)

March 30, 2002

time

Sample period, T

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Z-transform meaning of ‘z’

kzkTxzTxzTxzTxxtxZ )(....)3()2()()0()( 321

z-1 is the previous sample, or delay operator

z is the next or future sample, prediction operator

z is also the complex variable of the z-transform

The z-transform is similar to the Laplace transform

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Digital Control Systems:

The z-transform

0

k

f kT( ) zk

Z{f(t)} = F(z) =

Def’n of z-Transform:

Relationship between s-plane and z-plane: a delay

z esT

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Digital Control Systems:

The z-transform (cont) Several common z-Transforms

x(t) X(s) X(z)

________________________________________________

u t( )1

s

z

z 1

t1

s2

Tz

z 1( )2

eat 1

s a

z

z eaT

sin t

s2

2

z sin T

z2

2z cos T 1

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Digital Control Systems:

Stability in s-plane

Region of Stability:

Left-Hand Plane

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Digital Control Systems:

Stability in z-plane

Region of Stability:

Unit-Circle

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Digital Control Systems:

Root-Locus

Y z( )

R z( )

KG z( ) D z( )

1 KG z( ) D z( )

1 KG z( ) D z( ) 0

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TABLE 8.1 Laplace Transforms and z-Transforms of Simple

Discrete Time Functions

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TABLE 8.1 (continued) Laplace Transforms and z-Transforms of Simple

Discrete Time Functions

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TABLE 8.1 (continued) Laplace Transforms and z-Transforms of Simple

Discrete Time Functions

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Discrete Root Locus, LH plane of s-plane maps to a unit circle in

the z-plane

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Figure 8.4 Natural frequency (solid color) and damping loci (light color) in

the z-plane; the portion below the Re(z)-axis (not shown) is the mirror image of the upper half shown

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Figure 8.5 Time sequences associated with points in the z-plane

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Digital Control Systems:

MATLAB to the rescue

• c2d – conversion of continuous-time models to discrete time

• zgrid – generate z-plane grid lines for a root locus or pole-zero

map over an existing map

• dstep – Step response of discrete time system

• stairs – Connects the elements from dstep to form stairstep graph

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Digital Control Systems:

Design Example

Root Locus Design for Digital

DC Motor Position Control

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Digital Control Systems:

Design Example

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Comparison of (a) digital and; (b) continuous implementation

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Trapezoidal integration in Tustin’s method

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Simulink block diagram for transient response of lead-compensation designs with

discrete and analog implementations

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Comparison between the digital (using Tustin’s) and the continuous controller step response with a sample rate 25 times bandwidth: (a) position; (b)

control signal

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Comparison between the digital (using ZOH approximation) and the continuous controller step response with a sample

rate 25 times bandwidth: (a) position; (b) control

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As CPU’s get faster discrete systems approach continuous.

Digital stability check is always recommended

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Digital Control Systems: Sampling real

signals: Zero-Order Hold

H2 s( )1 e

sT

s