Low Parametric Sensitivity realization design for T. Hilaire, P. - PowerPoint PPT Presentation

CAO’06 — Low Parametric Sensitivity realization design for T. Hilaire, P. Chevrel, J.P. Clauzel FWL implementation of MIMO Controllers Introduction Theory and application to the active control of vehicle Low longitudinal oscillations Sensitivity Realizations Implicit State-Space Framework T. Hilaire 1 , 3 P. Chevrel 1 , 2 J.P. Clauzel 3 TF Sensitivity Measure 1 IRCCyN UMR CNRS 6597 NANTES FRANCE Optimal Design 2 ´ Ecole des Mines de Nantes NANTES FRANCE Conclusion 3 PSA Peugeot Citro¨ en LA GARENNE COLOMBES FRANCE CAO’06 - 26-28 April 2006 - Cachan France 1/32

Context CAO’06 — T. Hilaire, P. Chevrel, J.P. Clauzel Linear Time Invariant filters or controllers Introduction Finite Word Length implementation of control algorithms Low Sensitivity Realizations Implicit Motivation State-Space Framework Evaluate the impact of the quantization of the embedded TF Sensitivity coefficients Measure Compare various realizations and find an optimal one Optimal Design Conclusion 2/32

Context CAO’06 — T. Hilaire, P. Chevrel, J.P. Clauzel Linear Time Invariant filters or controllers Introduction Finite Word Length implementation of control algorithms Low Sensitivity Realizations Implicit Motivation State-Space Framework Evaluate the impact of the quantization of the embedded TF Sensitivity coefficients Measure Compare various realizations and find an optimal one Optimal Design Conclusion 2/32

Outline CAO’06 — T. Hilaire, P. Chevrel, The classical low sensitivity realization problem 1 J.P. Clauzel Introduction Macroscopic representation of algorithms through the 2 Low implicit state-space framework Sensitivity Realizations Implicit State-Space The transfer function sensitivity measure 3 Framework TF Sensitivity Measure The optimal realization design problem 4 Optimal Design Conclusion Conclusion and Perspectives 5 3/32

Outline CAO’06 — T. Hilaire, P. Chevrel, The classical low sensitivity realization problem 1 J.P. Clauzel Introduction Macroscopic representation of algorithms through the 2 Low implicit state-space framework Sensitivity Realizations Implicit State-Space The transfer function sensitivity measure 3 Framework TF Sensitivity Measure The optimal realization design problem 4 Optimal Design Conclusion Conclusion and Perspectives 5 4/32

FWL degradation CAO’06 — T. Hilaire, P. Chevrel, Origin of the degradation J.P. Clauzel The deterioration induced by the FWL implementation comes Introduction from : Low Sensitivity Quantization of the involved coefficients Realizations → parametric errors Implicit State-Space Framework Roundoff noises in numerical computations TF Sensitivity → numerical noises Measure Optimal Design Only the deterioration induced by the quantization of Conclusion coefficients is considered here. 5/32

FWL degradation CAO’06 — T. Hilaire, P. Chevrel, Origin of the degradation J.P. Clauzel The deterioration induced by the FWL implementation comes Introduction from : Low Sensitivity Quantization of the involved coefficients Realizations → parametric errors Implicit State-Space Framework Roundoff noises in numerical computations TF Sensitivity → numerical noises Measure Optimal Design Only the deterioration induced by the quantization of Conclusion coefficients is considered here. 5/32

Equivalent realizations CAO’06 — T. Hilaire, Let’s consider a transfer function H ( z ) and one of its P. Chevrel, J.P. Clauzel realization ( A q , B q , C q , D q ) Introduction H ( z ) = C q ( zI − A q ) − 1 B q + D q Low Sensitivity � qX k Realizations = A q X k + B q U k Implicit with qX k � X k +1 State-Space Y k = C q X k + D q U k Framework TF Sensitivity Measure The realizations of the form ( T − 1 A q T , T − 1 B q , C q T , D q ), with Optimal Design T a non-singular matrix, are all equivalent in infinite precision. Conclusion They are no more in finite precision. 6/32

Equivalent realizations CAO’06 — T. Hilaire, Let’s consider a transfer function H ( z ) and one of its P. Chevrel, J.P. Clauzel realization ( A q , B q , C q , D q ) Introduction H ( z ) = C q ( zI − A q ) − 1 B q + D q Low Sensitivity � qX k Realizations = A q X k + B q U k Implicit with qX k � X k +1 State-Space Y k = C q X k + D q U k Framework TF Sensitivity Measure The realizations of the form ( T − 1 A q T , T − 1 B q , C q T , D q ), with Optimal Design T a non-singular matrix, are all equivalent in infinite precision. Conclusion They are no more in finite precision. 6/32

Transfer function sensitivity measure CAO’06 — T. Hilaire, P. Chevrel, Gevers and Li (1993) have proposed a measure of the J.P. Clauzel sensitivity of the transfer function with respect to the Introduction coefficients A , B and C Low Sensitivity 2 2 2 � � � � � � Realizations ∂ H ∂ H ∂ H M L 2 � � � � � � � + + Implicit � � � � � � ∂ A ∂ B ∂ C State-Space � � � � � � 2 2 2 Framework The optimal design problem consists in finding TF Sensitivity Measure Optimal M L 2 ( T − 1 AT , T − 1 B , CT , D ) argmin Design T non singular Conclusion 7/32

Transfer function sensitivity measure CAO’06 — T. Hilaire, P. Chevrel, Gevers and Li (1993) have proposed a measure of the J.P. Clauzel sensitivity of the transfer function with respect to the Introduction coefficients A , B and C Low Sensitivity 2 2 2 � � � � � � Realizations ∂ H ∂ H ∂ H M L 2 � � � � � � � + + Implicit � � � � � � ∂ A ∂ B ∂ C State-Space � � � � � � 2 2 2 Framework The optimal design problem consists in finding TF Sensitivity Measure Optimal M L 2 ( T − 1 AT , T − 1 B , CT , D ) argmin Design T non singular Conclusion 7/32

Outline CAO’06 — T. Hilaire, P. Chevrel, The classical low sensitivity realization problem 1 J.P. Clauzel Introduction Macroscopic representation of algorithms through the 2 Low implicit state-space framework Sensitivity Realizations Implicit State-Space The transfer function sensitivity measure 3 Framework TF Sensitivity Measure The optimal realization design problem 4 Optimal Design Conclusion Conclusion and Perspectives 5 8/32

The need of a unifying framework CAO’06 — T. Hilaire, P. Chevrel, J.P. Clauzel Various implementation forms have to be taken into consideration Introduction Low shift-realizations Sensitivity Realizations δ -realizations Implicit State-Space observer-state-feedback Framework direct form I or II TF Sensitivity Measure cascade or parallel realizations Optimal Design etc... Conclusion 9/32

The need of a unifying framework CAO’06 — T. Hilaire, P. Chevrel, In order to encompass all these implementations, we have J.P. Clauzel proposed a specialized implicit state-space realization to be Introduction used as a unifying framework : Low Sensitivity Realizations Interests Implicit State-Space macroscopic description of a FWL implementation Framework more general than previous realizations TF Sensitivity Measure more realistic with regard to the parameterization Optimal Design directly linked to the in-line computations to be performed Conclusion 10/32

The need of a unifying framework CAO’06 — T. Hilaire, P. Chevrel, In order to encompass all these implementations, we have J.P. Clauzel proposed a specialized implicit state-space realization to be Introduction used as a unifying framework : Low Sensitivity Realizations Interests Implicit State-Space macroscopic description of a FWL implementation Framework more general than previous realizations TF Sensitivity Measure more realistic with regard to the parameterization Optimal Design directly linked to the in-line computations to be performed Conclusion 10/32

The need of a unifying framework CAO’06 — T. Hilaire, P. Chevrel, In order to encompass all these implementations, we have J.P. Clauzel proposed a specialized implicit state-space realization to be Introduction used as a unifying framework : Low Sensitivity Realizations Interests Implicit State-Space macroscopic description of a FWL implementation Framework more general than previous realizations TF Sensitivity Measure more realistic with regard to the parameterization Optimal Design directly linked to the in-line computations to be performed Conclusion 10/32

The need of a unifying framework CAO’06 — T. Hilaire, P. Chevrel, In order to encompass all these implementations, we have J.P. Clauzel proposed a specialized implicit state-space realization to be Introduction used as a unifying framework : Low Sensitivity Realizations Interests Implicit State-Space macroscopic description of a FWL implementation Framework more general than previous realizations TF Sensitivity Measure more realistic with regard to the parameterization Optimal Design directly linked to the in-line computations to be performed Conclusion 10/32