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Three-term controller Ziegler-Nichols tuning rules Summary Basics of Automation and Control I Lecture 13: PID controllers Paweł Malczyk Division of Theory of Machines and Robots Institute of Aeronautics and Applied Mechanics Faculty of Power and Aeronautical Engineering Warsaw University of Technology © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 1 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Outline 1 Three-term controller 2 Ziegler-Nichols tuning rules 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 2 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Three-term controller 1 Three-term controller Introduction Basic control functions P controller PI controller PD controller PID controller 2 Ziegler-Nichols tuning rules 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 3 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Introduction Fig. 1: Block diagram of a process with feedback controller 1 The three-term controller, i.e. proportional-integral-derivative (PID) controller, is a control loop feedback mechanism widely used in many industrial control systems. 2 PID controllers appear in many different forms: as stand-alone controllers, as part of hierarchical, distributed control systems and built into embedded components. 3 By tuning the three parameters in the PID controller algorithm, the controller can provide control action designed for specific process requirements (benign process dynamics, moderate performance). © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 4 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Basic control functions Fig. 2: Block diagram of feedback control system with ideal PID controller The input/output relation for an ideal PID controller*: [ ∫ ] 1 t de(t) u(t) = k e(t) + e(τ)dτ + T = p T d dt ∫i 0 (1) t de(t) = kpe(t) + ki e(τ)dτ + kd 0 dt e(t) = r(t) − y(t) – error signal, u(t) – control signal, kp – proportional gain, Ti – integral time, Td – derivative time, k = kp – integral gain, k = k T – derivative gain. i Ti d p d * Modified versions of the PID controller are used in practical applications. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 5 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Basic control functions Ideal PID controller: [ ∫ ] t 1 de(t) u(t) = kp e(t) + e(τ)dτ + T T d dt i 0 ( ) (2) ( ) → U s 1 C(s) = = kp 1 + + Tds E(s) Tis Fig. 3: Block diagram of ideal PID controller • P controller (Ti = ∞, Td = 0) C(s) = kp (3) • PI controller (Td = 0)( ) 1 C(s) = kp 1 + (4) Tis • PD controller (Ti = ∞) C(s) = kp(1 + Tds) (5) Fig. 4: PID interpretation → Gentle introduction to PID controllers: https://www.youtube.com/watch?v=4Y7zG48uHRo. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 6 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Mechanical system Ns N Fig. 5: Układ mechaniczny. Stałe fizyczne: m = 1 kg, b = 2 m , k = 5 m Goal: choose a control force f(t) (in N) such that the mass will stop at the desired position (say 1 m from equilibrium). Y(s) 1 Plant tranfer function G(s) = F(s) = ms2+bs+k . Reference signal (desired position): r(t) = yd · 1(t) (yd = 1 m). In steady-state the force f(t) need to balance the spring force, i.e.: f = k · yd · 1(t), then 1 yss = lim y(t) = lim sY(s) = lim sG(s)F(s) = lim sG(s)kyd = G(0)kyd = yd t→∞ s→0 s→0 s→0 s © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 7 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Mechanical system with disturbance Fig. 6: Mechanical system. Physical constants: m = 1 kg, b = 2 Ns , k = 5 N , and g = 10 m m m s2 Y(s) 1 Plant transfer function G(s) = F(s) = ms2+bs+k . Disturbance – gravity force: d(t) = −mg · 1(t). Reference signal (desired position): r(t) = yd · 1(t) (yd = 1 m). In steady-state the force f(t) need to balance the sum of spring force and the gravity force, i.e.: u = (k · yd + mg) · 1(t). Assumption: perfect knowledge of parameters and disturbance... © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 8 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Mechanical system with parametric uncertainty True values: m, b, k. Estimated params: m, b, k. Real parameters and disturbance signals are usually unknown. Fig. 7: Mechanical system with parametric uncertainty Control signal: u = (k · yd + mg) · 1(t). Input signal: f = (kyd + (m − m)g) · 1(t). Steady-state response: − 1 1 − yss = lim y(t) = lim sY(s) = lim sG(s)(kyd+(m m)g) = (kyd+(m m)g) t→∞ s→0 s→0 s k Error signal: 1 1 e = y − y = y − (ky + (m − m)g) = ((k − k)y + (m − m)g) ss ss ss d k d k d The greater the uncertainty is, the greater the steady-state error is. Transient-response is not controlled. Open-loop control is sensitive to parametric uncertainty. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 9 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Proportional control Fig. 8: Feedback control system Assumption: u(t) = kp(r(t) − y(t)) – force proportional to the error. Open-loop transfer function: L(s) = C(s)G(s). Y(s) L(s) kp Closed-loop transfer function: T(s) = = = 2 . R(s) 1+L(s) ms +bs+k+kp kp – virtual spring coefficient. Let us calculate the response y(t) and steady-state error e(t). © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 10 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Proportional control Fig. 9: Feedback control system The response: Y(s) = T(s)R(s) + G(s)S(s)D(s). Sensitivity function: 1 ms2 + bs + k ( ) = = S s 2 1 + L(s) ms + bs + k + kp Load disturbance sensitivity function: 1 ( ) = ( ) ( ) = W s G s S s 2 ms + bs + k + kp · → yd Reference signal: r(t) = yd 1(t) R(s) = s − · → − mg Disturbance: d(t) = mg 1(t) D(s) = s . © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 11 / 35 Three-term controller Ziegler-Nichols tuning rules Summary Proportional control Fig. 10: Feedback control system Steady-state response: yd mg yss = lim y(t) = lim sY(s) = lim s(T(s) + W(s)(− )) = t→∞ s→0 s→0 s s − kp − 1 = ydT(0) mgW(0) = yd mg k + kp k + kp If k → ∞, then kp → 1 and 1 → 0. In consequence: y = y . p k+kp k+kp ss d Conclusion: → disturbance rejection (even unknown) and trajectory tracking. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 12 / 35 Three-term controller Ziegler-Nichols tuning rules Summary P controller Fig. 11: P controller, C(s) = kp Effects of P controller Proportional term, P, causes a corrective control actuation proportional to the error. The system with P controller will usually have nonzero steady-state errors. As kp increases, then the static position error decreases. As kp increases, the stability margins decrease and the system may become unstable. As kp increases, the BW (as well as overshoot and settling time) increases. It is difficult to ensure both good transient response accuracy and steady- state performance. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 13 / 35 Three-term controller Ziegler-Nichols tuning rules Summary P controller kp=30 40 1.4 30 1.2 20 10 1 r(t) 0 0.8 -10 -20 y(t) 0.6 kp=30 kp=25 -30 0.4 kp=20 Magnitude (dB) -40 0.2 kp=15 -50 -60 0 10-2 10-1 100 101 102 0 1 2 3 4 5 6 7 8 9 10 Frequency (rad/sec) t kp=30 0 30 25 -30 20 -60 15 10 -90 G C u(t) 5 kp=15 -120 L 0 kp=20 Phase (deg) S -5 kp=25 -150 T -10 kp=30 -180 -15 10-2 10-1 100 101 102 0 1 2 3 4 5 6 7 8 9 10 Frequency (rad/sec) t Fig. 12: P controller characteristics Comment 1 If kp ↗, then Mp ↗, tp ↘, ts ↗, ess ↘, stability issues. © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 14 / 35 Three-term controller Ziegler-Nichols tuning rules Summary P controller kp=30 40 1.4 30 1.2 20 10 1 r(t) 0 0.8 -10 y(t) 0.6 -20 kp=30 kp=25 Magnitude (dB) G -30 0.4 kp=20 C -40 L kp=15 S 0.2 -50 T -60 0 10-2 10-1 100 101 102 0 1 2 3 4 5 6 7 8 9 10 Frequency (rad/sec) t Fig. 13: P controller characteristics © Paweł Malczyk. Basics of Automation and Control I Lecture 13: PID controllers 15 / 35 Three-term controller Ziegler-Nichols tuning rules Summary PI controller ( ) 1 Fig. 14: PI controller, C(s) = kp 1 + Tis Effects of PI controller The integral term, I, gives a correction proportional to the integral of the error. PI largely reduces the steady-state errors (compared to P controller). It may cause the closed loop system less stable (or even unstable). PI may slow down the transient response.
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