93 9393 932009-09-29 13:11:30 UTC / rev b19331f50bbd7Control7.1 Motor model with feedforward control 837.2 Simple feedback control 857.3 Sensor delays 877.4 Inertia 90The 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 thecontrol variable, angular velocity, is one derivative different from angle.We first make a discrete-time model of such a system and try to controlit without feedback. To solve the problems of the feedforward setup, wethen introduce feedback and analyze its effects.7.1 Motor model with feedforward controlWe would like to design a controller that tells the motor how to place thearm at a given position. The simplest controller is entirely feedforward inthat it does not use information about the actual angle. Then the high-levelblock diagram of the controller–motor system is94 9494 9484 7.1 Motor model with feedforward control2009-09-29 13:11:30 UTC / rev b19331f50bbdcontrollermotorinputoutputwhere 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 varywith time, as it would for a robot arm being directed toward a teacup (fora 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. Withthe output signal being the same kind of quantity as the input signal (bothare angles), a feedback controller can easily compute the error signal bysubtracting the output from the input.With this setup, the controller–motor system takes the desired angle as itsinput signal and produces the actual angle of the arm as its output.To design the controller, we need to model the motor. The motor turnsa voltage into the arm’s angular velocity ω. The continuous-time systemthat turns ω into angle is θ ∝Rω dt. Its forward-Euler approximation isthe difference equationy[n] = y[n − 1] + x[n − 1].The corresponding system functional is R/(1 − R), which represents anaccumulator 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 travelis not (yet?) available, so a bare R in a denominator, which represents anegative delay, is impossible. A realizable controller is 1 − R, which pro-duces a single delay R for the combined system functional:R1 − R1 − Rcontrollermotorinputoutput95 9595 957 Control 852009-09-29 13:11:30 UTC / rev b19331f50bbdAlas, the 1 − R controller is sensitive to the particulars of the motor and ofour model of it. Suppose that the arm starts with a non-zero angle beforethe motor turns on (for example, the whole system gets rotated without themotor knowing about it). Then the output angle remains incorrect by thisinitial angle. This situation is dangerous if the arm belongs to a 1500-kgrobot where an error of 10◦means that its arm crashes through a brick wallrather than stopping to pick up the teacup near the wall.A problem in the same category is an error in the constant of proportional-ity. Suppose that the motor model underestimates the conversion betweenvoltage and angular velocity, say by a factor of 1.5. Then the system func-tional of the controller–motor system is 1.5R rather than R. A 500-kg armmight again arrive at the far side of a brick wall.One remedy for these problems is feedback control, whose analysis is thesubject of the next sections.7.2 Simple feedback controlIn feedback control, the controller uses the output signal to decide whatto tell the motor. Knowing the input and output signals, an infinitely in-telligent controller could deduce how the motor works. Such a controllerwould realize that the arm’s angle starts with an offset or that the mo-tor’s conversion is incorrect by a factor of 1.5, and it would compensate forthose and other problems. That mythical controller is beyond the scope ofthis course (and maybe of all courses). In this course, we use only linear-systems theory rather than strong AI. But the essential and transferableidea in the mythical controller is feedback.So, sense the the angle of the arm, compare it to the desired angle, and usethe difference (the error signal) to decide the motor’s speed:+controllermotorsensor−1controllermotorsensorA real sensor cannot respond instantaneously, so assume the next-best sit-uation, that the sensor includes one unit of delay. Then the sensor’s outputgets subtracted from the desired angle to get the error signal, which is used96 9696 9686 7.2 Simple feedback control2009-09-29 13:11:30 UTC / rev b19331f50bbdby the controller. The simplest controller, which uses so-called propor-tional control, just multiplies the error signal by a constant β. This setuphas the block diagram+C(R) = βM(R) =R1 − RS(R) = R−1controllermotorsensorIt was analyzed in lecture and has the system functionalC(R)M(R)1 + C(R)M(R)S(R)=βR/(1 − R)1 + βR2/(1 − R).Multiply by (1 − R)/(1 − R) to clear the fractions. ThenF(R) =βR1 − R + βR2,where F(R) is the functional for the whole feedback system.Let’s analyze its behavior in the extreme cases of the gain β. As β → ∞,the system functional limits to R/R2= 1/R, which is a time machine. Sincewe cannot build a time machine just by choosing a huge gain in a feedbacksystem, some effect should prevent us raising β → ∞. Indeed, instabilitywill prevent it, as we will see by smoothly raising β from 0 to ∞.To study stability, look at the poles of the feedback system, which are givenby the factors of the denominator 1 − R + βR2. The factored form is (1 −p1R)(1 − p2R). So the sum of the poles is 1 and their product is β. At theβ → 0 extreme, which means no feedback, the poles are approximately 1−β and β. The pole near 1 means that the system is almost an accumulator; itapproximately integrates the input signal. This behavior is what the motordoes without feedback and
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