On Optimal Input Design in System Identification for Model - PowerPoint PPT Presentation

On Optimal Input Design in System Identification for Model Predictive Control Mariette Annergren Joint work with Christian A. Larsson and Håkan Hjalmarsson ACCESS and Automatic Control Lab KTH Royal Institute of Technology, Stockholm, Sweden

Outline Introduction Notation 1. Theory Application Set System 2. Identification Algorithm Identification Set Optimal Input 3. MPC Example Signal Design Identification Algorithm 4. Conclusions Identification Algorithm 5. Future Work Identification Algorithm MPC Example MPC Example MPC Example MPC Example 2 Conclusions

Introduction Definition Introduction Framework for experiment design in system identification Notation for control, specifically MPC. Application Set System Identification Set • Objective: Optimal Input Signal Design Find an input signal that minimizes the cost related to Identification the system identification experiment. Algorithm Identification Algorithm • Constraint: Identification Algorithm A specified control performance is guaranteed when MPC Example MPC Example using the estimated model in the control design. MPC Example MPC Example 3 Conclusions

Notation Introduction Notation Application Set The model structure is parametrized by θ . System Identification Set • True system is given by θ 0 . Optimal Input Signal Design • Estimated model is given by ˆ Identification θ . Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example 4 Conclusions

Application Set • Application cost: V app ( θ , θ 0 ) , for example T V app ( θ , θ 0 ) = 1 Introduction � y t ( θ 0 ) − y t ( θ ) � 2 ∑ 2 . T Notation t = 1 Application Set System Identification Set • Application specification: Optimal Input Signal Design V app ( θ , θ 0 ) ≤ 1 Identification γ , γ > 0 . Algorithm Identification Algorithm Identification Algorithm • Acceptable parameter set: MPC Example � θ | V app ( θ , θ 0 ) ≤ 1 � MPC Example Θ app ( γ ) = . MPC Example γ MPC Example 5 Conclusions

Application Set (cont.) Ellipsoidal approximation: Introduction � � app ( θ 0 , θ 0 )( θ − θ 0 ) ≤ 2 θ | ( θ − θ 0 ) T V ′′ Notation Θ app ( γ ) ≈ E app ( γ ) = . γ Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example 6 Conclusions

Application Set (cont.) Scenario approach: � � θ i , i = 1 ... M < ∞ | V app ( θ i , θ 0 ) ≤ 1 Θ app ( γ ) ≈ Introduction . γ Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example More on scenario approach: MPC Example MPC Example G. C. Calafiore and M. C. Campi, 2006. MPC Example 7 Conclusions

System Identification Set Introduction Notation Asymptotic quality property: Application Set System θ | ( θ − θ 0 ) T I F ( θ − θ 0 ) ≤ χ 2 � � Identification Set α ( n ) ˆ θ ∈ E SI ( α ) = . Optimal Input N Signal Design Identification (Key result from prediction error/maximum likelihood system Algorithm Identification identification.) Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example 8 Conclusions

Optimal Input Signal Design • Estimated parameters: Introduction θ | ( θ − θ 0 ) T I F ( θ − θ 0 ) ≤ χ 2 � � α ( n ) ˆ Notation θ ∈ E SI ( α ) = . N Application Set System Identification Set Optimal Input • Acceptable parameters in application: Signal Design Identification Algorithm � θ | V app ( θ , θ 0 ) ≤ 1 � ˆ θ ∈ Θ app ( γ ) = . Identification γ Algorithm Identification Algorithm MPC Example • Experiment cost: MPC Example f cost (Φ u ) . MPC Example MPC Example 9 Conclusions

Optimal Input Signal Design (cont.) Optimization Problem Introduction Notation minimize f cost (Φ u ) Application Set Φ u subject to E SI ( α ) ⊆ Θ app ( γ ) System Identification Set 0 ≤ Φ u ( ω ) , ∀ ω Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example 10 Conclusions

Optimal Input Signal Design (cont.) Optimization Problem Introduction Notation minimize f cost (Φ u ) Application Set Φ u subject to E SI ( α ) ⊆ Θ app ( γ ) System Identification Set 0 ≤ Φ u ( ω ) , ∀ ω Optimal Input Signal Design Identification Can be approximated as a convex problem! Algorithm Identification Algorithm Using: Identification • ellipsoidal approximation ⇒ LMI, Algorithm MPC Example • scenario approach ⇒ scalar linear inequalities, MPC Example • finite dimensional parametrization ⇒ LMI. MPC Example MPC Example 10 Conclusions

Identification Algorithm Issues: Introduction Notation • θ 0 is unknown. Application Set System Identification Set • Evaluation of V app ( θ , θ 0 ) . Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example 11 Conclusions

Identification Algorithm Issues: Introduction Notation • θ 0 is unknown. Application Set System Identification Set • Evaluation of V app ( θ , θ 0 ) . Optimal Input Signal Design Identification Solutions: Algorithm Identification • Use estimates instead of θ 0 . Algorithm Identification Algorithm • Evaluate V app ( θ , θ 0 ) in simulation. MPC Example MPC Example MPC Example MPC Example 11 Conclusions

Identification Algorithm (cont) Proposed algorithm: Introduction 1. Find an initial estimate of θ 0 . Notation Application Set 2. Evaluate V app ( θ , θ 0 ) in simulation. System Identification Set 3. Design the optimal input signal. Optimal Input Signal Design Identification 4. Find a new estimate of θ 0 . Algorithm Identification Algorithm Identification Algorithm Discussion on iterative approach: MPC Example L. Gerencsér, H. Hjalmarsson, J. Mårtensson, 2009. MPC Example MPC Example MPC Example 12 Conclusions

MPC Example Introduction Notation x 4 Application Set Control objective: System x 3 Identification Set Reference Optimal Input Signal Design γ 1 γ 2 tracking of the Identification Algorithm lower tank levels Identification using MPC. Algorithm x 1 x 2 Identification Algorithm u 1 u 2 MPC Example MPC Example MPC Example MPC Example 13 Conclusions

MPC Example (cont) • Application cost: Introduction Notation T V app ( θ , θ 0 ) = 1 Application Set � y t ( θ 0 ) − y t ( θ ) � 2 ∑ 2 . System T Identification Set t = 1 Optimal Input Signal Design Identification Algorithm • Experiment cost: Input power, Identification � 1 Algorithm � π � Identification f cost (Φ u ) = trace − π φ u ( ω ) d ω . Algorithm 2 π MPC Example MPC Example MPC Example MPC Example 14 Conclusions

MPC Example (cont) Simulation MPC Introduction u M (ˆ MPC( ˆ θ ) θ ) Notation y (ˆ θ , ˆ θ ) Application Set r System Identification Set y ( θ , ˆ θ ) Optimal Input u Signal Design M (ˆ MPC( θ ) θ ) Identification Algorithm Identification Algorithm Identification Algorithm MPC Example V app ( θ , θ 0 ) ≈ V app ( θ , ˆ θ ) = 1 T ∑ T t = 1 � y t (ˆ θ , ˆ θ ) − y t ( θ , ˆ θ ) � 2 MPC Example 2 MPC Example MPC Example 15 Conclusions

16 Water level [cm] 15 • Optimal: 91 % 14 Introduction success. Notation 100 150 200 250 300 Time [s] Application Set System Identification Set 16 Optimal Input Signal Design Water level [cm] • White: Identification Algorithm 15 15 % Identification Algorithm success. Identification Algorithm 14 MPC Example 100 150 200 250 300 MPC Example Time [s] MPC Example White: 8 × N gives same success rate as optimal. MPC Example 16 Conclusions

Conclusions Introduction Notation Application Set • Identification algorithm for MPC. System Identification Set • Increased control performance. Optimal Input Signal Design Identification Algorithm • Linear framework applicable on nonlinear systems. Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example 17 Conclusions

Future Work Introduction Notation • How to choose V app . Application Set System Identification Set • Realistic MPC applications. Optimal Input Signal Design • Closed-loop identification. Identification Algorithm Identification Algorithm • Toolbox for optimal input design, MOOSE. Identification Algorithm MPC Example MPC Example MPC Example MPC Example 18 Conclusions

MOOSE, www.ee.kth.se/moose Definition Introduction Notation MOOSE is a model based optimal input design toolbox Application Set developed for Matlab. System Identification Set Optimal Input It features Signal Design Identification Algorithm • optimal input design, Identification Algorithm • easy-to-use text interface, Identification Algorithm • compatibility with Matlab Control System Toolbox. MPC Example MPC Example MPC Example MPC Example 19 Conclusions