1. Design Realization lecture 23
John Canny
11/13/03

2. Last time Circuit design critique Control principles
Simulation – Matlab/Simulink

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

4. PD Control
In addition to position feedback, a multiple of the velocity (derivative) is fed back as well to stabilize the system:

5. 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.

6. 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.

7. PID Tracking Controller
All three terms P,I,D are computed on the difference signal:
PID controller

8. Example2: Pendubot
A two-axis robot. The first (blue) link is driven, the second (red) link is passive.
The model includes gravity, and is quite non-linear.

9. Example2: Pendubot
Use feedback and feedforward PID blocks to stabilize two oscillatory modes.

10. 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.

11. Discrete lowpass amplifier
Input is (x1,…,xn), output is (y1,…,yn)
yk = a yk-1 + (1-a)b xk a, b constants, a < 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.

12. Discrete lowpass amplifier
The amplifier’s DC Gain is b
Corner frequency c = (- ln a)/t = 2fc where t is the discrete step time.

13. 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.

14. 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.

15. Example: Real-time Theremin Model
1-wire sensors control pitch and volume.
Real-time sound output via sound card.

16. 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 Simulink’s timestamps to real time.

17. 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.

18. 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.

19. 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.

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