6 003 Signals and Systems CT Feedback and Control October 29 2009 Feedback and Control Continuous time feedback has many applications Examples improve performance of an op amp circuit control position of a motor reduce sensitivity to unwanted parameter variation reduce distortions stabilize unstable systems magnetic levitation inverted pendulum Feedback and Control Reducing sensitivity to unwanted parameter variation Example power amplifier power amplifier MP3 player F0 speaker 8 F0 12 Changes in F0 due to changes in temperature for example lead to undesired changes in sound level Feedback and Control Feedback can be used to compensate for parameter variation power amplifier MP3 player X F0 K 8 F0 12 H s KF0 1 KF0 If K is made large so that KF0 1 then 1 H s independent of K or F0 Y speaker Feedback and Control Feedback reduces the change in gain due to change in F0 MP3 player X F0 100 Y 8 F0 12 1 10 Gain to Speaker 20 F0 no feedback 100F0 feedback 0 1 100F 10 10 8 F0 12 0 F0 0 10 20 Check Yourself power amplifier MP3 player X F0 K 8 F0 12 Y speaker Feedback greatly reduces sensitivity to variations in K or F0 lim H s K KF0 1 1 KF0 What about variations in Aren t those important Check Yourself What about variations in Aren t those important The value of is typically determined with resistors whose values are quite stable compared to semiconductor devices Crossover Distortion Feedback can compensate for parameter variation even when the variation occurs rapidly Example using transistors to amplify power 50V MP3 player speaker 50V Crossover Distortion This circuit introduces crossover distortion For the upper transistor to conduct Vi Vo VT For the lower transistor to conduct Vi Vo VT Vo 50V Vi Vo VT VT 50V Vi Crossover Distortion Crossover distortion can have dramatic effects Example crossover distortion when the input is Vi t B sin 0 t Vo t 50V Vi Vo 50V t Crossover Distortion Feedback can reduce the effects of crossover distortion 50V MP3 player K speaker 50V Crossover Distortion When K is small feedback has little effect on crossover distortion 50V Vi Vo t Vo K 50V K 1 t Crossover Distortion As K increases feedback reduces crossover distortion 50V Vi Vo t Vo K 50V K 2 t Crossover Distortion As K increases feedback reduces crossover distortion 50V Vi Vo t Vo K 50V K 4 t Crossover Distortion As K increases feedback reduces crossover distortion 50V Vi Vo t Vo K 50V K 10 t Crossover Distortion original distorted K 2 K 4 K 8 K 16 original Crossover Distortion original distorted K 2 K 4 K 8 K 16 original Crossover Distortion original distorted K 2 K 4 K 8 K 16 original Crossover Distortion original distorted K 2 K 4 K 8 K 16 original Crossover Distortion original distorted K 2 K 4 K 8 K 16 original Crossover Distortion original distorted K 2 K 4 K 8 K 16 original Crossover Distortion original distorted K 2 K 4 K 8 K 16 original Crossover Distortion original distorted K 2 K 4 K 8 K 16 original Crossover Distortion original distorted K 2 K 4 K 8 K 16 original Feedback and Control Continuous time feedback has many applications Examples improve performance of an op amp circuit control position of a motor reduce sensitivity to unwanted parameter variation reduce distortions stabilize unstable systems magnetic levitation inverted pendulum Control of Unstable Systems Feedback is also useful for controlling unstable systems Example Magnetic levitation i t io y t Control of Unstable Systems Magnetic levitation is unstable i t io f t y t Mg Equilibrium y 0 magnetic force f t is equal to the weight M g Increase y greater force further increase y Decrease y reduced force further decrease y Positive feedback Control of Unstable Systems Magnetic levitation is unstable i t io f t y t Mg f t i t i0 Mg y t Control of Unstable Systems The instability of magnetic levitation is similar to that of a ball poised at the apex of perfectly smooth hill Levitation with a Spring By contrast the mass and spring system is not unstable F K x t y t M y t x t y t Levitation with a Spring If the body moves up or down the spring force decreases or increases and the body tends to fall back or rise up F K x t y t M y t f t Mg K y t Levitation with a Spring The spring and mass system is a stable system analogous to a ball in a valley Levitation with a Spring The block diagram for the spring and mass system has negative feedback because the slope of the force curve is negative F K x t y t M y t x t K M y t A y t A y t Levitation with a Spring This system is marginally stable poles on imaginary axis or stable poles in left half plane if effects of friction are included s plane 0 q K M q K 0 M Magnetic Levitation By contrast the system representing magnetic levitation has positive feedback f t i t i0 Mg y t x t p20 y t A y t A y t Magnetic Levitation The poles are at s p s plane p0 p0 Magnetic Levitation We can stabilize this system by adding an additional feedback loop to control i t f t i t 1 1i0 i t i0 i t 0 9i0 Mg y t Magnetic Levitation Magnetic levitation can be stabilized using a controller with both proportional plus derivative terms x t 1 s y t A s plane p0 p0 y t A y t Magnetic Levitation Without feedback one pole is in the right half plane unstable With sufficient feedback all poles are in left half plane stable s plane p0 p0 Here feedback is used to stabilize an otherwise unstable system Try it Demo designed by Prof James Roberge Inverted Pendulum As a final example of stabilizing an unstable system consider an inverted pendulum m t d2 x t dt2 mg t l mg l x t lab frame inertial cart frame non inertial 2 d2 x t 2 d t ml mg l sin t m l cos t z dt2 z z dt2 z z I force distance force distance Check Yourself Inverted Pendulum Where are the poles of this system t m mg d2 x t dt2 t l x t ml2 d2 x t d2 t mgl sin t m l cos t dt2 dt2 mg l Check Yourself Inverted Pendulum As a final example of stabilizing an unstable system consider an inverted pendulum t m mg d2 x t dt2 t l x t ml2 d2 t d2 x t mgl sin t m l cos t dt2 dt2 ml2 d2 t d2 x t mgl t ml dt2 dt2 s2 l mls2 2 H s 2 2 X ml s mgl s g l mg l r g poles at …
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