# chap6 - 93 93 7 Control 7.1 7.2 7.3 7.4 Motor model with...

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7 Control 7.1 Motor model with feedforward control 83 7.2 Simple feedback control 85 7.3 Sensor delays 87 7.4 Inertia 90 The goals of this chapter are to study: how to use feedback to control a system; how slow sensors destabilize a feedback system; and how to model inertia and how it destabilizes a feedback system. A common engineering design problem is to control a system that inte- grates. For example, position a rod attached to a motor that turns input (control) voltage into angular velocity. The goal is an angle whereas the control variable, angular velocity, is one derivative different from angle. We ﬁrst make a discrete-time model of such a system and try to control it without feedback. To solve the problems of the feedforward setup, we then introduce feedback and analyze its effects. 7.1 Motor model with feedforward control We would like to design a controller that tells the motor how to place the arm at a given position. The simplest controller is entirely feedforward in that it does not use information about the actual angle. Then the high-level block diagram of the controller–motor system is

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84 7.1 Motor model with feedforward control controller motor input output where we have to ﬁgure out what the output and input signals represent. A useful input signal is the desired angle of the arm. This angle may vary with time, as it would for a robot arm being directed toward a teacup (for a robot that enjoys teatime). The output signal should be the variable that interests us: the position (angle) of the arm. That choice helps later when we analyze feedback con- trollers, which use the output signal to decide what to tell the motor. With the output signal being the same kind of quantity as the input signal (both are angles), a feedback controller can easily compute the error signal by subtracting the output from the input. With this setup, the controller–motor system takes the desired angle as its input signal and produces the actual angle of the arm as its output. To design the controller, we need to model the motor. The motor turns a voltage into the arm’s angular velocity ω . The continuous-time system that turns ω into angle is θ R ωdt . Its forward-Euler approximation is the difference equation y [ n ] = y [ n 1 ] + x [ n 1 ] . The corresponding system functional is R / ( 1 R ) , which represents an accumulator with a delay. Exercise 37. Draw the corresponding block diagram. The ideal output signal would be a copy of the input signal, and the cor- responding system functional would be 1 . Since the motor’s system func- tional is R / ( 1 R ) , the controller’s should be ( 1 R ) / R . Sadly, time travel is not (yet?) available, so a bare R in a denominator, which represents a negative delay, is impossible. A realizable controller is 1 R , which pro- duces a single delay R for the combined system functional: R 1 R 1 R controller motor input output
7 Control 85 Alas, the 1 R

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chap6 - 93 93 7 Control 7.1 7.2 7.3 7.4 Motor model with...

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