Design Realization lecture 22

1. Design Realization lecture 22 John Canny 11/6/03

2. Last time • Some physics: • Bending and stretching • Construction methods: • Molding • Welding • Structural components • Modular systems

3. This time • Circuit design critique • Control principles • Simulation – Matlab/Simulink

4. Feedback Control • Often we want to move a system in a particular way, by controlling a parameter such as: • Position control • Speed control • Force control • Feedback control uses sensor(s) to measure this parameter and make corrections. • Feedback must be applied with care to avoid ever-increasing corrections (instability).

5. - + Feedback Control • Naively, we want to do something like this: V+ Amplifier Input(voltagerepresentingdesired angle) Potentiometer on shaft(angle sensor) Motor

6. - + Feedback Control • Any difference between input and measured shaft angle will be amplified, moving the motor. • If the direction is correct, the motor will reduce the difference. With high gain, the error  0. V+ Amplifier Input Potentiometer on shaft(angle sensor) Motor

7. Simulink Models • Tools like Matlab/Simulink allow us to design and test controllers before building them. • Here is the controller just shown in Simulink: Voltage angle

8. Feedback Instability • Problem: the amplifier has delay, the motor has inertia, keeps moving even after error  0. • If gain is too high, it will overshoot, “ring” or possibly oscillate.

9. PD Stabilizing Controller • The simplest way to control feedback is with a “PD” (Proportional Derivative) controller. • A multiple of the derivative of the output is subtracted from the amplifier input.

10. 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.

11. 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.

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

13. Implementing PID Controllers • Normally, the controller CPU is running at fixed discrete time steps. • Derivates can be computed by differencing consecutive samples, integrals by summing samples. • 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.

14. 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.

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

16. 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.

17. 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.