Design Realization lecture 23

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Design Realization lecture 23

• John Canny
• 11/13/03

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Last time

• Circuit design critique
• Control principles
• Simulation Matlab/Simulink

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This time

• Finish circuit design critique
• Graphical programming and real-time control
  (Simulink).
• Automatic real-time code generation (Real-Time
  Workshop).

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PD Control

• In addition to position feedback, a multiple of
  the velocity (derivative) is fed back as well to
  stabilize the system

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PD Stabilization

• Why does derivative feedback stabilize the
  system?
• Derivative feedback simulates a damper.
• Motion in a fluid creates viscous drag (F ? -v).
• Viscous drag quickly robs the system of energy.

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PID Control

• Sometimes there is a residual error between
  desired and actual output (not for DC motors).
• Computing the integral of the difference signal
  will reduce it to zero in the steady state.

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PID Tracking Controller

• All three terms P,I,D are computed on the
  difference signal

PID controller
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Example2 Pendubot

• A two-axis robot. The first (blue) link is
  driven, the second (red) link is passive.
• The model includesgravity, and is
  quitenon-linear.

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Example2 Pendubot

• Use feedback and feedforward PID blocks to
  stabilize two oscillatory modes.

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Implementing PID Controllers

• Normally, the controller CPU is running at fixed
  discrete time steps.
• Derivatives can be computed by differencing
  consecutive samples, integrals by summing samples
  since time zero.
• This approach introduces delays and can cause
  problems at high frequency.
• Make sure that amplifiers roll off at high
  frequency use a low-pass amplifier.

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Discrete lowpass amplifier

• Input is (x1,,xn), output is (y1,,yn)
• yk a yk-1 (1-a)b xk a, b
  constants, a lt 1.
• If x 0, y non-zero, then the amplifier outputs
  a decreasing geometric sequence, which is a
  discrete approximation to exponential decay.
• It simulates a simple RC low-pass circuit.

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Discrete lowpass amplifier

• The amplifiers DC Gain is b
• Corner frequency ?c (- ln a)/t 2?fcwhere t
  is the discrete step time.

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Transfer functions

• The variable s represents frequency
• T(s) 1/s is an integrator
• T(s) s is a derivative operator
• T(s) 1/(a bs) is a low-pass filter with
  corner frequency a/b
• T(s) (c ds) is a high-pass filter with corner
  frequency c/d
• T(s) (a bs)/(c ds) is a general gain block,
  DC gain a/c, high frequency gain b/d.

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Real-time Simulink execution

• Simulink does not have a real-time clock (at
  least under Windows), and runs in virtual time.
• It can be driven in real-time if one of its
  blocks (especially input/output blocks) updates
  at regular real-time speed.
• The remainder of the Simulink code must run fast
  enough to keep up.

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Example Real-time Theremin Model

• 1-wire sensors control pitch and volume.
• Real-time sound output via sound card.

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Example Real-time Simulink Model

• The simsound block includes a separate thread
  which sends data to the sound card.
• This thread signals the main program thread with
  a semaphore when it is ready to accept data, or
  when its buffer is full.
• The buffer contains data with time stamps at the
  desired real-time update rate.
• Simulink runs a little faster than the sound card
  update rate, and the sound thread interpolates
  from Simulinks timestamps to real time.

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Example Real-time Simulink Model

• The AtoD block also includes a separate thread to
  acquire A-to-D input data.
• This thread runs as fast as it can to provide the
  most up-to-date sensor data to the running
  simulation.

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Automatic code generation

• There is a companion to Matlab/Simulink called
  real-time workshop (RTW).
• RTW automatically generates C code to run a
  Simulink model. It can handle new user-defined
  blocks (e.g. for sensor input or motor output).
• This code can be compiled and run on the control
  processor.

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Automatic code generation

• RTW code generation includes scheduling and
  event-handling and allows blocks to run at
  different rates.
• It also allows complicated models that may not
  run correctly with a simple discrete-step
  approximation.

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Summary

• Feedback control methods PD and PID.
• Feedforward control.
• Real-time use of Simulink.
• Code generation with Real-Time Workshop.