Design Realization
lecture 22
John Canny
11/6/03
Last time
 Some physics:
 Bending and stretching
 Construction methods:




Molding
Welding
Structural components
Modular systems
This time
 Circuit design critique
 Control principles
 Simulation – Matlab/Simulink
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).
Feedback Control
 Naively, we want to do something like this:
-
V+
Amplifier
+
Input
(voltage
representing
desired angle)
Motor
Potentiometer on shaft
(angle sensor)
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
Motor
Potentiometer on shaft
(angle sensor)
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
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.
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.
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.
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.
PID Tracking Controller
 All three terms P,I,D are computed on the
difference signal:
PID controller
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.
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.
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.
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.
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.