PI Tuning via Extremum Seeking Methods for Cruise Control

Transcription

1 PI Tuning via Extremum Seeking Methods for Cruise Control Yiyao(Andy) ) Chang Scott Moura ME 569 Control of Advanced Powertrain Systems Professor Anna Stefanopoulou December 6, 27 Yiyao(Andy) Chang and Scott Moura Slide 1 of 17

2 Motivation Automated PID Tuning Method Online control synthesis Achieve optimal performance in some sense Reduces calibration time Does not require knowledge of plant Does not require special experiments Can ES be used as an adaptive control law? Real-time controller adaptation Varying plant parameters and dynamics Varying disturbance inputs Yiyao(Andy) Chang and Scott Moura Slide 2 of 17

3 Reference Paper PID Tuning Using Extremum Seeking Online, Model-Free Performance Optimization Nick J. Killingsworth and Miroslav Krstic IEEE Control Systems Magazine February 26 Yiyao(Andy) Chang and Scott Moura Slide 3 of 17

4 Literature Review Closed-loop (model free) PID parameter tuning Relay Feedback Tuning Åström et al (1984) Leva (1993) Voda et al (1995) Unfalsified Control Jun et al (1999) Saeki et al (23, 24) Iterative Feedback Tuning Hjalmarsson et al (1998) Lequin et al (1999, 23) Yiyao(Andy) Chang and Scott Moura Slide 4 of 17

5 Outline Motivation & Literature Review Paper Reproduction Extremum Seeking (ES) Algorithm PID Tuning on Sample Plant ES Parameter Sensitivity Case Study: Adaptive Cruise Control Cruise Control Model PI Tuning Vary Desired Velocity, Fix Road Grade Vary Road Grade, Fix Desired Velocity Vahidi Road Grade Profile (I-15 near San Diego) Summary & Conclusions Yiyao(Andy) Chang and Scott Moura Slide 5 of 17

6 Extremum Seeking Algorithm 1.) 2.) 3.) 4.) 5.) ES Initialize Evaluate Estimate Perturb algorithm PID new cost parameters gains PID gradient gains θ J h ω i α i γ θˆ PID parameters Cost Washout filter cutoff freq Perturbation freq. Perturbation amplitude Adaptation gain PID parameters estimate Yiyao(Andy) Chang and Scott Moura Slide 6 of 17

7 Control Problem Formulation Cost Function: 1 T 2 J( θ) = e ( t, θ) dt T t t et, θ = rt yt, θ ( ) ( ) ( ) Plant: 1 = 1 + 2s 2 ( ) e G s Controller: Cr ( s) = K 1+ 1 Ts i 1 Cy( s) = K 1+ + Tds Ts i Tuning Parameters: θ = [ K T T ] T i d s Yiyao(Andy) Chang and Scott Moura Slide 7 of 17

8 PID Tuning on Sample Plant Use Ziegler-Nichols as initial gains X: 1.4 Y: 1.33 X: Y:.127 K 1.2 Cost X: 2.8 Y: X: 2 Y: 5.21e Iterations (k) 1.5 Ti X: Y: 31 X: 2 Y: y(t).5 8 X: Y: 7.74 ZN IMC IFT ES Time (sec) Td X: Y: Iterations (k) Yiyao(Andy) Chang and Scott Moura Slide 8 of 17

9 PID Gain Trajectories wrt Cost 8.5 x Td Ti Cross-section of cost as a function of T d and T i Fixed K Yiyao(Andy) Chang and Scott Moura Slide 9 of 17

10 Varying ES Parameters How sensitive are the results to the choice of the ES parameters? 1.6 x α,γ α/2,γ α,γ/1 α/2,γ/1 Recall α i γ Perturbation amplitude Adaptation gain Cost J(θ) Stable Region Unstable Region Iterations (k) Killingsworth s Answer fairly insensitive Our Answer Insensitive for more conservative values VERY sensitive for more aggressive values Yiyao(Andy) Chang and Scott Moura Slide 1 of 17

11 Case Study Adaptive Cruise Control Yiyao(Andy) Chang and Scott Moura Slide 11 of 17

12 ĵ ˆk F ˆ xi î Vehicle Dynamics Model ( cosθ ˆ sinθ ˆ) mg j + i ˆ Nj F iˆ drag ma iˆ = F mgsinθ fn F x θ ma ˆj = N mgcosθ = ˆ fni drag Assumptions Rolling disk dynamics No wheel slip No actuator dynamics Coulomb friction ( ) 2 F =.5ρ AC v + v N drag d w = mgcosθ dv m = F mgsinθ fmgcosθ.5ρac v+ v dt ( ) 2 x d w Wind Speed Yiyao(Andy) Chang and Scott Moura Slide 12 of 17

13 Linearization and Ziegler-Nichols δ v = ρacd ( v + vw) δv+ δfx + [ mgcosθ + fmgsinθ] δθ m m m Variable Nominal Value.8 Step Response Traction Force, F x Road Grade, θ 293 N radians.7.6 System: sys Rise Time (sec): 167 System: sys Final Value:.758 Wind Speed, v w Vehicle Speed, v ( ) ( ) 2 m/s 2 m/s 45 mph Y s.758 K = = U s 75.75s+ 1 τ s+ 1 PI Controller: C s ( ) K PI τ PI s + 1 s Amplitude Time (sec) = ( ) ( ) 43 s + 1 Ziegler-Nichols C s.3845 Yiyao(Andy) Chang and Scott Moura Slide 13 of 17 = Open Loop Step Response s

14 PI Tuning for Cruise Control Speed Error[m/s] Traction Force [N] Control Methods Extremum Seeking (ES) Fixed Ziegler-Nichols Gains (ZN Fix) Gain Scheduling (ZN GS) ES ZN Fix ZN GS ES ES Filtered ZN Fix ZN GS Time (sec) Actual Speed [m/s] Time (sec).7 ES.6 ES Filtered ZN Fix.5 ZN GS PI Gain: K pi PI Gain: τ PI.4.3 ES ZN Fix ZN GS Desired ES 36 ZN Fix ZN GS Iterations (k) Yiyao(Andy) Chang and Scott Moura Slide 14 of 17

15 Summary & Conclusions Tunes PID controllers by minimizing a cost function characterizing the desired closed-loop behavior Achieves better or comparable results relative to other popular tuning methods Improper ES parameters may produce instability Successfully performs adaptation for time-varying systems Online adaptation does not require models Yiyao(Andy) Chang and Scott Moura Slide 15 of 17

16 References [1] N.J. Killingsworth, M. Krstic, PID tuning using extremum seeking, IEEE Control Systems Magazine, pp 7-79, Feb. 26. [2] K.J. Åström, B. Wittenmark, Computer Controlled Systems: Theory and Design, 3 rd ed. Upper Saddle River, NJ: Prentice-Hall, [3] O. Lequin, E. Bosmans, T. Triest, Iterative feedback tuning of PID parameters: Comparison with classical tuning rules, Contr. Eng. Pract., vol. 11, no. 9, pp , 23. [4] A. Vahidi, A. Stefanopoulou, H. Peng, Recursive least squares with forgetting for online estimation of vehicle mass and road grade: theory and experiments, International Journal of Vehicle Mechanics and Mobility, Jan 25. Yiyao(Andy) Chang and Scott Moura Slide 16 of 17

17 QUESTIONS? Yiyao(Andy) Chang and Scott Moura Slide 17 of 17

18 APPENDIX SLIDES Yiyao(Andy) Chang and Scott Moura

19 Results: Road Grade Steps Road Grade [rad] PI Gain: K pi ES ES Filter ZN Actual Speed [m/s] 2 PI Gain: τ PI Traction Force [N] Cost Distance (m) Iterations (k) Yiyao(Andy) Chang and Scott Moura

20 Results: Vahidi Road Grade Profile (Linear Model) Cost ES ZN Road Grade [rad] PI Gain: K pi Actual Speed [m/s] PI Gain: τ PI Traction Force [N] Iterations (k) Distance (m) Yiyao(Andy) Chang and Scott Moura