digital control systems -...
TRANSCRIPT
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Digital Control
Systems
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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