 TODO add: PID material from Pont slides Some inverted pendulum videos Model-based control and other more sophisticated controllers?controllers? More code speed issues – perf with and w/o FP on different processorsLast Time Estimating worst case execution time Holistic scheduling: real-time guarantees over a CAN busToday Feedback (closed loop) control and PID controllers Principles Implementation TuningControl Systems Definitions: Desired state variables: X*(t)• These are given to the system from outside• E.g. “room temperature should be 70°” Estimated state variables: X’(t)•E.g. “room temperature is currently 67°”•E.g. “room temperature is currently 67°”• Estimation uses standard data acquisition techniques– E.g. thermistor + ADC Actions: U(t)• System commands U(t) converted into driving forces using transducers• E.g. “turn off the furnace”More Control Terms Goal of a control system is to minimize error:  e(t) = | X*(t) – X’(t) | Evaluating a control system Steady-state controller error• Average e(t)Transient responseTransient response• How long the system takes to reach 99% of the desired final value after X*(t) is changed Stability• Does the system reach a steady state (smooth constant output) or does it oscillate?Early Feedback Control Centrifugal governor for a throttleSimple Closed Loop Example We’re designing a thermostat Goal is to heat room to a set temperature We can turn the furnace on and off When furnace is off, room cools down by itself Fancy control is not needed here!Rather we just need two temperature settingsRather we just need two temperature settings Tlow– temperature at which we turn on the furnace Thigh– temperature at which we turn off the furnace The difference between these provides hysteresis An on/off (“bang bang”) controller works well when the physical system is slow to respond What happens if control loop runs too fast or slow?Thermostat Codeint Thigh, Tlow;bool on = false;timer_interrupt_handler (void) {int T = read_adc();if (T < Tlow && !on) {turn_furnace_on();on = true;}if (T > Thigh && on) {turn_furnace_off();on = false;}}Another Simple Controller We’re controlling a robot arm attached to a stepper motor Need to get the arm to a certain position Position sensor tells us where the arm is Algorithm:if E > 1mm then decrement position by 1mmif E > 1mm then decrement position by 1mm if E < -1mm then increment position by 1mm Pretty tough to go wrong with a strategy like this But slow to converge What happens if control loop runs too fast or slow?PID Controllers Simple controllers like previous two examples work but in many real situations Respond slowly Have large errors Better answer: PID (Proportional Integral Derivative)PID equation:PID equation: KP, KI, and KDare design parameters Can be fixed at zero to simplify the controllerdttdEKdEKtEKtUtDIP)()()()(0∫++=ττPID Intuition KP– determines response to current error Too large → Oscillation Too small → Slow response Can’t eliminate steady-state error KI– determines response to cumulative errorCan eliminate steady-state error but often makes transient Can eliminate steady-state error but often makes transient response worse KD– determines response to rate of change of error Damps response when error is moving in the right direction Amplifies response when error is moving in the wrong direction Can be used to increase stability and improve transient responsePID Loop Skeleton Codedouble UpdatePID (SPid *pid, double error, double position) { …}}position = ReadPlantADC(); drive = UpdatePID (&plantPID, plantCommand - position,position);DrivePlantDAC (drive);Proportional Controldouble UpdatePID (SPid *pid, double error, double position) { double pTerm;double pTerm;pTerm = pid->pGain * error;return pTerm;}Example System 1Example System 2 Problem: Proportional control can’t always eliminate steady-state errorExample System 3 Problem: When there is too much actuator delay, proportional control cannot stabilize the systemIntegral Controldouble iTerm;pid->iState += error;if (pid->iState > pid->iMax) { pid->iState = pid->iMax;} else if (pid->iState < pid-> iMin) { } else if (pid->iState < pid-> iMin) { pid->iState = pid->iMin;}iTerm = pid->iGain * iState;  In practice, always used in conjunction with proportional controlExample 2 with PI Control Steady state error is eliminatedIntegrator Issues Sampling time becomes more important Sampling time shouldn’t vary more than 1%-5% On average, sampling should be periodic Integrator range is important “Integrator windup” occurs when the integrator value gets stuck too high or too lowstuck too high or too low Limiting the range makes this less of a problem Reasonable heuristic: Set integrator limits to drive limits Usually, if you can’t stabilize a system with proportional control, you can’t stabilize it with PI control eitherDifferential Controldouble dTerm;dTerm = pid->dGain * (position - pid->dState);pid->dState = position; Basically just computes the slope of the system state Attempts to predict system future based on recent past historyExample 3 with PD ControlDifferential Issues Derivative is very sensitive to noise In practice, might keep a buffer of several samples into the past in order to smooth out noise• This is just a low-pass filter• Decreases responsiveness – which is the point of differential control in the first placedifferential control in the first place Important for sampling period to be very even Do sampling in a high-priority interrupt or threadFull PID Source Codetypedef struct {double dState; // Last position inputdouble iState; // Integrator statedouble iMax, iMin; double iGain, // integral gainpGain, // proportional gaindGain; // derivative gain} SPid;double UpdatePID (SPid * pid, double error, double position) {double pTerm, dTerm, iTerm;pTerm = pid->pGain * error; pid->iState += error;if (pid->iState > pid->iMax) pid->iState = pid->iMax;else if (pid->iState < pid->iMin) pid->iState = pid->iMin;iTerm = pid->iGain * iState;dTerm = pid->dGain * (position - pid->dState);pid->dState = position;return pTerm + iTerm - dTerm;}Implementation Issues Example code uses floating point Convert to integer or