Closed-loop System, PID Controller
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1 Closed-loop System, PID Controller M. Fikar Department of Information Engineering and Process Control Institute of Information Engineering, Automation and Mathematics FCFT STU in Bratislava TAR MF (IRP) Closed-loop System, PID Controller WS 207/208 / 44
2 Contents Closed-loop System On-Off Controller PID Controller Description of Components Structures Practical Aspects MF (IRP) Closed-loop System, PID Controller WS 207/208 2 / 44
3 Closed-loop System Block Scheme of the Closed-loop System MF (IRP) Closed-loop System, PID Controller WS 207/208 4 / 44
4 Closed-loop System Closed-loop Transfer Functions Tracking problem: w y Regulation problem: r y G yw = Y(s) W(s) = G yr = Y(s) R(s) = G p G a G R +G m G p G a G R G pr +G m G p G a G R Characteristic equation of the closed-loop system +G m G p G a G R = 0 MF (IRP) Closed-loop System, PID Controller WS 207/208 5 / 44
5 Closed-loop System Simplified Block Scheme Tracking error: e(t) = w(t) y(t) Permanent (steady-state) tracking error: e( ) = w( ) y( ) MF (IRP) Closed-loop System, PID Controller WS 207/208 6 / 44
6 Closed-loop System Closed-loop Transfer Functions Tracking problem: w y Regulation problem: r y G yw = Y(s) W(s) = G SG R +G S G R G yr = Y(s) R(s) = G pr +G S G R Characteristic equation of the closed-loop system Open-loop transfer function +G S G R = 0 G OLTF = G S G R MF (IRP) Closed-loop System, PID Controller WS 207/208 7 / 44
7 On-Off Controller On-Off Controller u(t) = { umax if e(t) > 0 u min if e(t) < 0 u u max e u min MF (IRP) Closed-loop System, PID Controller WS 207/208 9 / 44
8 Proportional Controller u(t) = Z R e(t) PID Controller Description of Components u Z R Proportionality band: t u max u min = Z R P p, Z R = 00 P p Small P p : small offset, large oscillations. Large P p : large offset, smooth response. Steady-state error compensation: u(t) = Z R e(t)+u b (t) MF (IRP) Closed-loop System, PID Controller WS 207/208 / 44
9 Description of Components P Controller: First Order System / Disturbance Rejection Problem Problem definition: G S = Z Ts +, G R = Z R, G pr = What is the value of steady-state tracking error? y( ) = lim s 0 sy(s), Y(s) = G yr R(s) Z pr G pr T pr s+ G yr = = +G S G R + Z Ts+ Z R Z pr (Ts + ) y( ) = lim s 0 s = (T pr s + )(Ts + +ZZ R ) Z pr T pr s +, r(t) = A Z pr (Ts + ) (T pr s + )(Ts + +ZZ R ) A s = Z pr +ZZ R A 0 e( ) = w( ) y( ) = 0 Z pr +ZZ R A = Z pr +ZZ R A MF (IRP) Closed-loop System, PID Controller WS 207/208 2 / 44
10 Description of Components P Controller: First Order System / Setpoint Tracking Problem Problem definition: G S = Z Ts +, G R = Z R, w(t) = A What is the value of steady-state tracking error? y( ) = lim s 0 sy(s), Y(s) = G yw W(s) G yw = G SG R +G S G R = y( ) = lim s 0 s Z Ts+ Z R + Z Ts+ Z R ZZ R = Ts + +ZZ R ZZ R A Ts + +ZZ R s = ZZ R A < A +ZZ R e( ) = w( ) y( ) = A ZZ R +ZZ R A = +ZZ R A MF (IRP) Closed-loop System, PID Controller WS 207/208 3 / 44
11 Description of Components Proportional Controller - Simulations First order system Z R = Z R =3 Z R = w Z R = Z R =3 Z R =0 y/az y/a t/t Disturbance rejection t/t Setpoint tracking MF (IRP) Closed-loop System, PID Controller WS 207/208 4 / 44
12 Description of Components Proportional Controller - Simulations Higher order system various gains Z R w,y 0.5 w u t MF (IRP) Closed-loop System, PID Controller WS 207/208 5 / 44
13 Description of Components Integral Controller u(t) = e(t)dt T I G R (s) = T I s 0 u T I t MF (IRP) Closed-loop System, PID Controller WS 207/208 6 / 44
14 Description of Components PI Controller Improvement of P controller with varying u b U b (s) = Properties of PI controller: Includes past information T I s + U(s), U(s) = Z R Increases order of the closed-loop system Eliminates steady-state error Reduces speed of control T I s + E(s) T I s Destabilises the closed-loop (adds unstable pole, stable zero) MF (IRP) Closed-loop System, PID Controller WS 207/208 7 / 44
15 Description of Components PI Controller: First Order System / Disturbance Rejection Problem Problem definition: G S = Z Ts +, G R = Z R What is the value of steady-state tracking error? y( ) = lim sy(s), Y(s) = G yr R(s) s 0 Z pr (Ts + )s G yr = (T pr s + )(Ts 2 + s + ZZ R s + ZZ R ( + ), G pr = Z pr T I s T pr s +, r(t) = A T I ) Z pr (Ts + )s A y( ) = lim s s 0 (T pr s + )(Ts 2 + s + ZZ R s + ZZ R T I ) s = 0 A = 0 ZZ R T I e( ) = w( ) y( ) = 0 0 = 0 MF (IRP) Closed-loop System, PID Controller WS 207/208 8 / 44
16 Description of Components PI Controller: First Order System / Setpoint Tracking Problem Problem definition: G S = Z Ts +, G R = Z R What is the value of steady-state tracking error? ( + ), w(t) = A T I s y( ) = lim s 0 sy(s), Y(s) = G yw W(s) G yw = G SG R +G S G R = y( ) = lim s 0 s Z(Z R s + Z R TI ) Ts 2 + s + Z(Z R s + Z R TI ) Z(Z R s + Z R TI ) Ts 2 + s + Z(Z R s + Z R TI ) e( ) = w( ) y( ) = A A = 0 A s = A MF (IRP) Closed-loop System, PID Controller WS 207/208 9 / 44
17 Description of Components PI Controller - Simulations First order system T i =2 T i = T i = w T i =2 T i = T i =0.2 y/az y/a t/t Disturbance rejection t/t Setpoint tracking MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
18 Description of Components PI Controller - Simulations Higher order system various T I w,y.5 w u t MF (IRP) Closed-loop System, PID Controller WS 207/208 2 / 44
19 Description of Components Derivative Controller de(t) u(t) = T D dt de(t) e(t + T D ) e(t)+t D dt G R (s) = T Ds + T D N s Properties of D controller: Includes prediction of future Does not increase order of the closed-loop system Does not eliminate steady-state error Improves speed of control PD: adds one stable zero to the closed-loop u t MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
20 Description of Components PD Controller - Simulations Higher order system various T D w,y w u t MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
21 Structures Structures Without interaction ( R(s) = Z R + ) T I s + T Ds With interaction (in series) ( R(s) = Z R + ) (+T D s) T I s Parallel R(s) = Z R + T I s + T Ds MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
22 Structures Structures PI-D: no derivative kick ( U(s) = Z R + ) ( W(s) Z R + ) T I s T I s + T Ds Y(s) I-PD: no derivative and setpoint kick ( U(s) = Z R T I s W(s) Z R + ) T I s + T Ds Y(s) 2DoF PID: b, c <, (b (0.3, 0.8)) ( U(s) = Z R b + ) ( T I s + ct Ds W(s) Z R + ) T I s + T Ds Y(s) Honeywell: PID-A (Full PID), PID-B (PI-D), PID-C (I-PD) MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
23 Structures 2DoF Controller Separation into feedback and feedforward parts: ( R fb = Z R + ) ( T I s + T Ds, R ff = Z R b + ) T I s + ct Ds w R ff G(s) y R fb MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
24 Practical Aspects Integrator Windup Situation: controller with integral action and active constraints. u (0.9,.): 2.5 (a) w y u integ (b) w y u integ t MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
25 Practical Aspects Integrator Windup Solution Use original P+I controller with u b as feedback: u = Z R e+u b, u b = T I s + u SAT The signal u b uses u SAT after the saturation block. MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
26 Practical Aspects Integrator Windup Solution 2: Backintegration PI: T t T I, PID: T t T I T D Other possibilities: setpoint limitation, incremental implementation, tracking signal. MF (IRP) Closed-loop System, PID Controller WS 207/208 3 / 44
27 Practical Aspects AW: Linear Example Linearised two tank system with 2 states at q s 0 = 0.9, x 2 is measured. PI controller: Z r (+/T i s), Z r =, T i = MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
28 Practical Aspects Linear Example Simulink Scheme MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
29 Practical Aspects Linear Example Simulation Results h l,h-w 2 q time time MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
30 Practical Aspects Nonlinear Example Nonlinear two tank system with 2 states at q s 0 = 0.9, x 2 is measured with some noise. PI controller: Z r (+/T i s), Z r =, T i = MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
31 Practical Aspects Nonlinear Example Simulink Scheme MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
32 Practical Aspects Nonlinear Example Simulation Results h n,h-w q time time MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
33 Practical Aspects Nonlinear Example Simulink Scheme AW MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
34 Practical Aspects Nonlinear Example Simulation Results AW h n,h-w q time time MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
35 Practical Aspects Nonlinear Example Simulink Scheme AW2 MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
36 Practical Aspects Nonlinear Example Simulation Results AW h n,h-w q time time MF (IRP) Closed-loop System, PID Controller WS 207/208 4 / 44
37 Practical Aspects Bumpless Transfer Transfer between: Manual and automatic mode Several controllers (heating/cooling) Solutions: Incremental controllers Anti-windup schemes MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
38 Practical Aspects Other Issues Deadband No overshoot from known disturbances Noise filtering Derivative kick Controller parameter changes MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
39 Practical Aspects Digital Implementation PID changes to PSD (proportional, summation, derivative) Difference equation: u(k) = q 0 e(k)+q e(k )+q 2 e(k 2)+u(k ) Discrete time transfer function: R(z ) = q 0 + q z + q 2 z 2 z MF (IRP) Closed-loop System, PID Controller WS 207/ / 44
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