93 93 7 Control 7 1 7 2 7 3 7 4 Motor model with feedforward control Simple feedback control Sensor delays Inertia 83 85 87 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 integrates 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 93 2009 09 29 13 11 30 UTC rev b19331f50bbd 93 94 94 7 1 Motor model with feedforward control 84 input motor controller 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 controllers 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 R velocity The continuous time system that turns into angle is dt 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 corresponding system functional would be 1 Since the motor s system functional 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 produces a single delay R for the combined system functional input 1 R controller 94 R 1 R motor output 2009 09 29 13 11 30 UTC rev b19331f50bbd 94 95 95 7 Control 85 Alas the 1 R controller is sensitive to the particulars of the motor and of our model of it Suppose that the arm starts with a non zero angle before the motor turns on for example the whole system gets rotated without the motor knowing about it Then the output angle remains incorrect by this initial angle This situation is dangerous if the arm belongs to a 1500 kg robot where an error of 10 means that its arm crashes through a brick wall rather than stopping to pick up the teacup near the wall A problem in the same category is an error in the constant of proportionality Suppose that the motor model underestimates the conversion between voltage and angular velocity say by a factor of 1 5 Then the system functional of the controller motor system is 1 5R rather than R A 500 kg arm might again arrive at the far side of a brick wall One remedy for these problems is feedback control whose analysis is the subject of the next sections 7 2 Simple feedback control In feedback control the controller uses the output signal to decide what to tell the motor Knowing the input and output signals an infinitely intelligent controller could deduce how the motor works Such a controller would realize that the arm s angle starts with an offset or that the motor s conversion is incorrect by a factor of 1 5 and it would compensate for those and other problems That mythical controller is beyond the scope of this course and maybe of all courses In this course we use only linearsystems theory rather than strong AI But the essential and transferable idea in the mythical controller is feedback So sense the the angle of the arm compare it to the desired angle and use the difference the error signal to decide the motor s speed controller controller 1 motor motor sensor sensor A real sensor cannot respond instantaneously so assume the next best situation that the sensor includes one unit of delay Then the sensor s output gets subtracted from the desired angle to get the error signal which is used 95 2009 09 29 13 11 30 UTC rev b19331f50bbd 95 96 96 7 2 Simple feedback control 86 by the controller The simplest controller which uses so called proportional control just multiplies the error signal by a constant This setup has the block diagram controller C R 1 motor M R R 1 R S R R sensor It was analyzed in lecture and has the system functional R 1 R C R M R 1 C R M R S R 1 R2 1 R Multiply by 1 R 1 R to clear the fractions Then F R R 1 R R2 where F R is the functional for the whole feedback system Let s analyze its behavior in the extreme cases of the gain As the system functional limits to R R2 1 R which is a time machine Since we cannot build a time machine just by choosing a huge gain in a feedback system some effect should prevent us raising Indeed instability will prevent it as we will see by smoothly raising from 0 to To study stability look at the poles of the feedback system which are given by the factors of the denominator 1 R R2 The factored form is 1 p1 R 1 p2 R So the sum of the poles is 1 and their product is At the 0 extreme which means no feedback the poles are approximately 1 and The pole near 1 means that the system is almost an accumulator it approximately integrates the input signal This behavior is what the motor does without feedback and is far from our goal that the controller motor system copy the input to the output Turning up the gain improves the control However at the extreme the poles are roughly at p1 …
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