TODO add: PID material from Pont slides Some inverted pendulum videos Model-based control and other more sophisticated

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Samson George
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1 TODO add: PID material from Pont slides Some inverted pendulum videos Model-based control and other more sophisticated controllers? More code speed issues perf with and w/o FP on different processors

2 Last Time Estimating worst case execution time Holistic scheduling: real-time guarantees over a CAN bus

3 Today Feedback (closed loop) control and PID controllers Principles Implementation Tuning

4 Control 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 Estimation uses standard data acquisition techniques Actions: U(t) E.g. thermistor + ADC System commands U(t) converted into driving forces using transducers E.g. turn off the furnace

5 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 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?

6 Early Feedback Control Centrifugal governor for a throttle

7 Simple 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 T low temperature at which we turn on the furnace T high 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?

8 Thermostat Code int 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; } }

9 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 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?

10 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: U ( t) = KPE( t) + KI E( τ ) dτ + t 0 K D de( t) dt K P, K I, and K D are design parameters Can be fixed at zero to simplify the controller

11 PID Intuition K P determines response to current error Too large Oscillation Too small Slow response Can t eliminate steady-state error K I determines response to cumulative error Can eliminate steady-state error but often makes transient response worse K D 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 response

12 PID Loop Skeleton Code double UpdatePID (SPid *pid, double error, double position) { } position = ReadPlantADC(); drive = UpdatePID (&plantpid, plantcommand - position, position); DrivePlantDAC (drive);

13 Proportional Control double UpdatePID (SPid *pid, { } double pterm; double error, double position) pterm = pid->pgain * error; return pterm;

14 Example System 1

15 Example System 2 Problem: Proportional control can t always eliminate steady-state error

16 Example System 3 Problem: When there is too much actuator delay, proportional control cannot stabilize the system

17 Integral Control double iterm; 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; In practice, always used in conjunction with proportional control

18 Example 2 with PI Control Steady state error is eliminated

19 Integrator 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 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 either

20 Differential Control double 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 history

21 Example 3 with PD Control

22 Differential 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 place Important for sampling period to be very even Do sampling in a high-priority interrupt or thread

23 Full PID Source Code typedef struct { double dstate; // Last position input double istate; // Integrator state double imax, imin; double igain, // integral gain pgain, // proportional gain dgain; // 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; }

24 Implementation Issues Example code uses floating point Convert to integer or fixed-point by hand OR Keep using FP if the control loop frequency is low or if HW FP is available Multiplies can be avoided by selecting constants that are multiples of 2 Sampling rate Too low Slow response time Too high Differential noise and integral overflow Rule of thumb: Set sampling rate between 1/10 and 1/100 of desired settling time Difficult control problems argue for higher sampling rates

25 Tuning the Parameters Method 1: Plug system model into Matlab, it hands you the parameters Method 2: Tune by hand Happy fact: When the system is not too sensitive, control parameters only have to be roughly correct to do a good job Requirements for hand tuning: Need to be able to observe controller variables and output Need to be able to supply square-wave inputs Note: Parameters can interact in complex ways Tuning is an art Tuning by hand will be difficult if system is barely controllable System may be uncontrollable

26 Tuning Differential Gain Start here unless you can do without dgain, in which case set it to 0.0 and move to pgain Start with small proportional gain, e.g. 1.0 Start with dgain == pgain * 100 Increase dgain until unacceptable oscillation, overshoot, or noise occurs Oscillation from too much dgain is much faster than oscillation from not enough Back off by factor of 2-4 System will now be sluggish time to tune P and I

27 Tuning pgain and igain Start with a pgain around By experimentation find the value of pgain where oscillation starts Then back off by a factor of 2-4 Start with igain around Again find a value that gives reasonably good performance without causing oscillation

28 Conclusions Feedback control is a broadly useful technology for getting an embedded system to have some desired effect on the real world In practice manual tuning replaces analytic solutions However, Matlab has great support for feedback control Can generate efficient code too

1. Consider the closed loop system shown in the figure below. Select the appropriate option to implement the system shown in dotted lines using

1. Consider the closed loop system shown in the figure below. Select the appropriate option to implement the system shown in dotted lines using op-amps a. b. c. d. Solution: b) Explanation: The dotted