PID control - Simple tuning methods

- Simple tuning methods Ulf Holmberg

Introduction Lab processes Control System Dynamical System Step response model Self-oscillation model PID structure Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols) Experiment Level control in a tank Level control of two connected tanks

Tank process Tank with level control Two connected tanks Pump for in-flow of water Level measurements Valve for out-flow (disturbance)

Open-loop system System u Actuator Control object Sensor y u control signal (pump voltage) Actuator (tubes to pump, power amplifier, in-flow) Control object (level in Tank with in- and out-flow) Sensors (pressure sensors, tubes, elektronics) y output (measurement of water level)

Closed-loop system r + e Controller u System y Give reference (set-point of water level in tank) r to controller in stead of pump voltage

Time constant and stationary gain u(t) 1 Step K p y(t) Step response t u System y T t Example Stove plate u(t) power to stove plate y(t) temperature on plate Time constant T Stationary gain K p

Time constant and stationary gain u u(t) Step K p u y(t) Step response t u System y T t Step response size proportional to step size Start step from equilibrium

Introduction Dynamical System Experiment Dead-time (time delay) u(t) y (t) Step response Kp Step 1 t u System y t L T Example roll transport time Dead-time (time delay, lag) L

Step response model u(t) 1 Step K p y(t) Step response t u System y A t Step response model from unit step: T Time constant K p Stationary gain L Dead-time A see measure above L T

Step response model u u(t) Step K p u y(t) Step response t u System y A u t Step response model from experiment: T Time constant K p Stationary gain L Dead-time A see figure L T

Self-oscillation model r + e K u u System y T u Experiment P-control (closed-loop system!) Crank up gain to self-oscillation Reference-step starts oscillation Self-oscillation model Ultimate gain K u Ultimate period T u

PID structure r + e Controller u System y r reference, set-point (SP) y output, measured signal to be regulated e = r y control error PID-controller: P e I + u D

P-controller e(t) u(t) Ke 0 e 0 t e K u t Control signal Proportional to control error

P-controller with offset e(t) u(t) Ke 0 e 0 t u 0 e K + u u 0 Offset t Control signal Proportional to error plus offset Example: speed control of car u 0 given gas at control-start

I-part (integrator) e(t) I-del e 0 e 0 t e 1 T i e I-del T i t I-part surface under e(t) so far becomes as big as constant error e 0 in time T i used to eliminate remaining error

D-part (derivator) e(t) D-del t e T d de dt D-del t D-part slope on e(t) used to damp oscillations

PID-controller structure Control signal = P + I + D u(t) = K [e(t) + 1 t de(t) e(s)ds + T d ] T i dt Three parameters to tune K, T i and T d

Ziegler-Nichols step response method Measure A and L from step response T p expected time constant for closed-loop system Controller K T i T d T p P 1/A 4L PI 0.9/A 3L 5.7L PID 1.2/A 2L L/2 3.4L

Ziegler-Nichols self-oscillation method Measure ultimate gain K u and period T u from experiment T p expected time constant for closed-loop system Controller K T i T d T p P 0.5K u T u PI 0.4K u 0.8T u 1.4T u PID 0.6K u 0.5T u 0.125T u 0.85T u

Step response Step response model for the left tank 10 5.4 9 5.3 8 7 6 5 4 3 5.2 5.1 5 4.9 4.8 4.7 4.6 Estimated from figure: A u = 0.1, u = 0.5 A = 1/5 L = 3 2 0 100 200 300 4.5 20 30 40 50 60

P-control of the left tank 3 r, y Valve opens/closes 2.5 0 20 40 60 80 100 120 140 160 180 u 4 Ziegler-Nichols P-control K = 1/A = 5 Stationary error after valve opening even if offset is used 2 0 0 20 40 60 80 100 120 140 160 180

PI-control of the left tank r, y Valve opens/closes 3 2.5 0 50 100 150 200 u 4 2 Ziegler-Nichols PI-control K = 0.9/A = 4.5 T i = 3L = 9 No stationary error after valve disturbance 0 0 50 100 150 200

PID-control of the left tank Valve opens/closes r, y 3.2 3 2.8 2.6 2.4 0 50 100 150 200 u 5 0 50 100 150 200 Ziegler-Nichols PID-control K = 1.2/A = 6 T i = 2L = 6 T d = L/2 = 1.5 No stationary error after valve disturbance Fast and damped Noisier control signal

Self-oscillation experiment Self-oscillation model for the left tank r (bˆ rvˆ rde) och y (ˆ rvˆ rde) 3 2 1 0 0 20 40 60 80 100 u (styrsignal) 10 5 Ultimate gain and period K u = 15 (from tuning) T u = 10 (from figure) Z-N PID-tuning: K = 0.6K u = 8 T i = 0.5T u = 5 T d = 0.125T u = 1.25 0 0 20 40 60 80 100 120

Ziegler-Nichols PID from self-oscillation model t, y 5 0 0 50 100 150 200 250 u 10 5 0 0 50 100 150 200 250

Step response Step response model for the right tank 8 7 6 5 4 3 2 0 100 200 300 400 4.7 4.6 4.5 4.4 4.3 4.2 4.1 4 3.9 30 40 50 60 70 Roughly estimated (bad precision!): A u 0.2, u = 0.5 A = 2/5 L 10 From larger figure: A u = 0.15, L = 8

PI-control of the right tank r, y Valve opens/closes 3 2 0 100 200 300 u 3 2 Ziegler-Nichols PI-control K = 0.9/A = 2.25 T i = 3L = 30 Disturbance eliminated in 60s (Compare T p = 5.7L = 57s) 1 0 0 100 200 300

PID-control of the right tank (step response tuning) Valve opens/closes r, y 6 5 4 3 0 100 200 300 u 10 5 Ziegler-Nichols PID-control K = 1.2/A = 3 T i = 2L = 20 T d = L/2 = 5 Disturbance eliminated i 40s (Compare T p = 3.4L = 34s) 0 0 100 200 300

Self-oscillation experiment y Self-oscillation model for the right tank 4.9 4.8 4.7 0 20 40 60 80 100 u 3 2 1 Ultimate gain and period K u = 10 (from tuning) T u = 25 (from figure) Z-N PID-tuning: K = 0.6K u = 6 T i = 0.5T u 13 T d = 0.125T u 3 0 0 20 40 60 80 100

PID-control of the right tank (self-oscillation tuning) 5 4 r, y Valve opens/closes 3 u 10 0 50 100 150 200 250 5 0 0 50 100 150 200 250