6 003 Signals and Systems Lecture 4 6 003 Signals and Systems September 22 2009 Feedback and Control Feedback is pervasive in natural and artificial systems Feedback and Control V p Turn steering wheel to stay centered in the lane desired position actual position car driver September 22 2009 Feedback and Control Feedback and Control Feedback is useful for regulating a system s behavior as when a thermostat regulates the temperature of a house Concentration of glucose in blood is highly regulated and remains nearly constant despite episodic ingestion and use glucose digestive glucose circulatory insulin system system food glucose cells tissues insulin pancreas cells desired temperature thermostat heating system glucose in actual temperature circulatory system stored glucose cells tissues insulin glucose concentration pancreas cells Feedback and Control Today s goal Motor control relies on feedback from pressure sensors in the skin as well as proprioceptors in muscles tendons and joints Use systems theory to gain insight into how to control a system Try building a robotic hand to unscrew a lightbulb Shadow Dexterous Robot Hand Wikipedia 1 6 003 Signals and Systems Lecture 4 September 22 2009 Example Steering a Car Outline of the lecture Algorithm steer left when car is right of center and vice versa Understanding the structure of a control problem automatic control feedback straight ahead steer right Analyzing feedback systems feedback cyclic paths persistent outputs steer right steer right Designing control systems constructing well behaved response properties straight ahead steer left steer left Bad algorithm poor performance Here we get persistent oscillations Structure of a Control Problem Three Parts in the Car Steering Problem plant car Simple Control systems have three parts X E C perfect sensor proportional controller Y controller plant X E C car controller S sensor S Y plant 1 sensor The plant is the system to be controlled The sensor measures the output of the plant The controller specifies a command C to the plant based on the difference between the input X and sensor output S X desired left right position in lane 0 C steering wheel position C E X S X Y Y actual left right position in lane Car Steering Problem Simpler Car Steering Problem A realistic model for the car is complicated Start with a simpler problem Imagine a very non physical car that can instantly change its lateral position from y n to y n c n 1 shift 128 X E C controller S car 1 shift 64 Y 1 shift 32 plant X 1 shift 16 1 E C controller shift 18 sensor S shift 14 shift y 0 Turning the steering wheel rotates the car body so that forward motion of the vehicle changes the position in the lane 1 What is the value of in this example What would be a better value of 2 1 sensor 12 car plant Y 6 003 Signals and Systems Lecture 4 Check Yourself Check Yourself On each step the car changes its position from y n to y n c n Y for the entire controller What is the system functional X system 1 shift 32 1 shift 16 shift shift 12 E X shift 18 14 C controller car Y 1 shift 32 plant 1 shift 16 S 1 y 0 September 22 2009 sensor 1 E C car controller Y plant shift 14 S shift 12 Find the appropriate model for this very unusual car 1 R X shift 18 1 y sensor 0 1 R R R 1 2 3 1 R 1 R 1 1 R 5 none of the above 1 1 R 3 4 R 1 R 1 R 5 none of the above 2 Designing a Controller Designing a Controller The closed loop system has a single pole at 1 Range of responses y n given y 0 1 and x n 0 R Y X 1 1 R 4 R 1 R y n 0 1 n 1 shift 32 1 shift 16 X E shift 18 C 12 controller y n car Y 12 plant n shift 14 y n S shift 12 1 y 0 1 sensor 1 n y n Therefore the output can deviate from the input for long periods of time as we saw when 21 32 n Designing a Controller Designing a Controller Long term deviations of the output from the input represent a failure to control the system in the desired fashion The open loop system car has a pole at z 1 an accumulator The closed loop system with feedback has a pole at z 1 Im We would like the car to instantly move to the desired position 1 shift 32 1 shift 16 X shift 18 E C controller car Y Re plant shift 14 S shift 12 y 0 1 1 sensor We can adjust to optimize performance Fortunately we can control the shape of the output Y R X 1 1 R Y R X 1 1 R by controlling the closed loop pole at 1 3 6 003 Signals and Systems Lecture 4 September 22 2009 Designing a Controller Modeling Sensor Delay The open loop system car has a pole at z 1 an accumulator The system that we just analyzed is idealized The closed loop system with feedback has a pole at z 1 Im Real systems may deviate from this analysis in important ways X E C 1 controller R 1 R plant Y Re S R sensor For example what would happen if the sensor introduced a delay We can adjust to optimize performance Y R X 1 1 R The value with the fastest response results when 1 Modeling Sensor Delay Check Yourself What would happen if the sensor introduced a delay X E see previous 0 command 0 C controller see previous 1 command 1 S see previous 1 command 1 R 1 R plant Y R sensor see previous 0 command 0 Which is true see previous 1 command 1 Introducing delay can destabilize a control system Key issue in biological and artificial control systems 1 Y R X 1 R 2 Y R X 1 R R2 3 Y R X 1 R R2 4 Y R R X 1 R 5 none of the above Modeling Sensor Delay Modeling Sensor Delay Find the closed loop poles If is small the fundamental modes occur at z and z 1 1 1 4 1 1 2 1 z 2 2 Substitute R z1 in the system functional z1 Y R z 2 X 1 R R2 z z 1 z1 12 0 z The poles are at 1 1 4 z 2 mode 1 Im z z plane n 0 1 2 3 4 1 Re z mode 2 n 0 1 2 3 4 Little feedback i e small slow decay of fundamental mode 2 slow system response 4 6 003 Signals and Systems Lecture 4 September 22 2009 Modeling Sensor Delay Modeling Sensor Delay As increases the fundamental modes move toward each other and collide at z 12 when 14 1 1 1 1 1 1 1 4 z 2 2 2 2 If 1 …
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