1Lecture 4:Basic Concepts in ControlCS 344R/393R: RoboticsBenjamin KuipersControlling a Simple System• Consider a simple system:– Scalar variables x and u, not vectors x and u.– Assume x is observable: y = G(x) = x– Assume effect of motor command u:• The setpoint xset is the desired value.– The controller responds to error: e = x − xset• The goal is to set u to reach e = 0.! ˙ x = F(x,u)! "F"u> 02The intuition behind control• Use action u to push back toward error e = 0– error e depends on state x (via sensors y)• What does pushing back do?– Depends on the structure of the system– Velocity versus acceleration control• How much should we push back?– What does the magnitude of u depend on?Velocity or acceleration control?• If error reflects x, does u affect x′ or x′′ ?• Velocity control: u → x′ (valve fills tank)– let x = (x)• Acceleration control: u → x′′ (rocket)– let x = (x v)T! ˙ x = (˙ x ) = F (x, u) = (u)! ˙ x =˙ x ˙ v " # $ $ % & ' ' = F (x, u) =vu" # $ $ % & ' ' ! ˙ v =˙ ˙ x = u3Laws of Motion in Physics• Newton’s Law: F=ma or a=F/m.• But Aristotle said:– Velocity, not acceleration, is proportional to theforce on a body.• Who is right? Why should we care?– (We’ll come back to this.)! ˙ x =˙ x ˙ v " # $ % & ' =vF / m" # $ % & ' The Bang-Bang Controller• Push back, against the direction of the error– with constant action u• Error is e = x - xset• To prevent chatter around e = 0,• Household thermostat. Not very subtle.! e < 0 " u := on " ˙ x = F (x, on) > 0e > 0 " u := off " ˙ x = F (x, off ) < 0! e < "#$ u := one > +#$ u := off4Bang-Bang Control in Action– Optimal for reaching the setpoint– Not very good for staying near itProportional Control• Push back, proportional to the error.– set ub so that• For a linear system, we get exponentialconvergence.• The controller gain k determines howquickly the system responds to error.! u = "ke + ub! ˙ x = F(xset,ub) = 0! x(t) = Ce"#t+ xset5Velocity Control• You want to drive your car at velocity vset.• You issue the motor command u = posaccel• You observe velocity vobs.• Define a first-order controller:– k is the controller gain.! u = "k (vobs" vset) + ubProportional Control in Action– Increasing gain approaches setpoint faster– Can leads to overshoot, and even instability– Steady-state offset6Steady-State Offset• Suppose we have continuing disturbances:• The P-controller cannot stabilize at e = 0.– Why not?! ˙ x = F(x,u) + dSteady-State Offset• Suppose we have continuing disturbances:• The P-controller cannot stabilize at e = 0.– if ub is defined so F(xset,ub) = 0– then F(xset,ub) + d ≠ 0, so the system changes• Must adapt ub to different disturbances d.! ˙ x = F(x,u) + d7Adaptive Control• Sometimes one controller isn’t enough.• We need controllers at different time scales.• This can eliminate steady-state offset.– Why?! u = "kPe + ub! ˙ u b= "kIe where kI<< kPAdaptive Control• Sometimes one controller isn’t enough.• We need controllers at different time scales.• This can eliminate steady-state offset.– Because the slower controller adapts ub.! u = "kPe + ub! ˙ u b= "kIe where kI<< kP8Integral Control• The adaptive controller means• Therefore• The Proportional-Integral (PI) Controller.! ˙ u b= "kIe! ub(t) = "kIe dt0t#+ ub! u(t) = "kPe(t) " kIe dt0t#+ ubNonlinear P-control• Generalize proportional control to• Nonlinear control laws have advantages– f has vertical asymptote: bounded error e– f has horizontal asymptote: bounded effort u– Possible to converge in finite time.– Nonlinearity allows more kinds of composition.! u = " f (e) + ubwhere f # M0+9Stopping Controller• Desired stopping point: x=0.– Current position: x– Distance to obstacle:• Simple P-controller:• Finite stopping time for! d = | x | +"! v =˙ x = " f (x)! f (x) = k | x | sgn(x)Derivative Control• Damping friction is a force opposingmotion, proportional to velocity.• Try to prevent overshoot by dampingcontroller response.• Estimating a derivative from measurementsis fragile, and amplifies noise.! u = "kPe " kD˙ e10Derivative Control in Action– Damping fights oscillation and overshoot– But it’s vulnerable to noiseEffect of Derivative Control– Different amounts of damping (without noise)11Derivative Control Can Add Noise– Why?Derivatives Amplify Noise– This is a problem if control output (CO)depends on slope (with a high gain).12The PID Controller• A weighted combination of Proportional,Integral, and Derivative terms.• The PID controller is the workhorse of thecontrol industry. Tuning is non-trivial.– Next lecture includes some tuning methods.! u(t) = "kPe(t) " kIe dt0t#" kD˙ e (t)PID Control in Action– But, good behavior depends on good tuning!– More on this later.13Exploring PI Control TuningHabituation• Integral control adapts the bias term ub.• Habituation adapts the setpoint xset.– It prevents situations where too much controlaction would be dangerous.• Both adaptations reduce steady-state error.! u = "kPe + ub! ˙ x set= +khe where kh<<kP14Types of Controllers• Feedback control– Sense error, determine control response.• Feedforward control– Sense disturbance, predict resulting error,respond to predicted error before it happens.• Model-predictive control– Plan trajectory to reach goal.– Take first step.– Repeat.Laws of Motion in Physics• Newton’s Law: F=ma or a=F/m.• But Aristotle said:– Velocity, not acceleration, is proportional to theforce on a body.• Who is right? Why should we care?! ˙ x =˙ x ˙ v " # $ % & ' =vF / m" # $ % & '15Who is right? Aristotle!• Try it! It takes constant force to keep anobject moving at constant velocity.– Ignore brief transients• Aristotle was a genius to recognize thatthere could be laws of motion, and toformulate a useful and accurate one.• This law is true because our everyday worldis friction-dominated.Who is right? Newton!• Newton’s genius was to recognize that thetrue laws of motion may be different fromwhat we usually observe on earth.• For the planets in orbit, without friction,motion continues without force.• For Aristotle, “force” means Fexternal.• For Newton, “force” means Ftotal.– On Earth, you must include Ffriction.16From Newton back to Aristotle• Ftotal = Fexternal + Ffriction•