ROCOND’06 — T. Hilaire, Pole Sensitivity Stability Related Measure of P. Chevrel, J.P. Clauzel FWL Realizations with the Implicit State-Space Introduction Formalism A pole sensitivity stability related measure T. Hilaire 1 , 3 P. Chevrel 1 , 2 J.P. Clauzel 3 Implicit State-Space Framework 1 IRCCyN UMR CNRS 6597 NANTES FRANCE Pole Sensitivity 2 ´ Ecole des Mines de Nantes NANTES FRANCE Measure Optimal 3 PSA Peugeot Citro¨ en LA GARENNE COLOMBES FRANCE realization Conclusion ROCOND’06 - 5-7 July 2006 - Toulouse France 1/31
Context ROCOND’06 — T. Hilaire, P. Chevrel, J.P. Clauzel Implementation of Linear Time Invariant controllers Introduction Finite Word Length context A pole sensitivity stability related Motivation measure Implicit Evaluate the impact of the quantization of the embedded State-Space Framework coefficients Pole Compare various realizations and find an optimal one Sensitivity Measure Optimal realization Conclusion 2/31
Context ROCOND’06 — T. Hilaire, P. Chevrel, J.P. Clauzel Implementation of Linear Time Invariant controllers Introduction Finite Word Length context A pole sensitivity stability related Motivation measure Implicit Evaluate the impact of the quantization of the embedded State-Space Framework coefficients Pole Compare various realizations and find an optimal one Sensitivity Measure Optimal realization Conclusion 2/31
Outline ROCOND’06 — T. Hilaire, P. Chevrel, A pole sensitivity stability related measure 1 J.P. Clauzel Introduction Macroscopic representation of algorithms through the 2 A pole implicit state-space framework sensitivity stability related measure Extension of the pole-sensitivity stability related measure 3 Implicit State-Space Framework Pole Optimal realization 4 Sensitivity Measure Optimal Conclusion and Perspectives 5 realization Conclusion 3/31
Outline ROCOND’06 — T. Hilaire, P. Chevrel, A pole sensitivity stability related measure 1 J.P. Clauzel Introduction Macroscopic representation of algorithms through the 2 A pole implicit state-space framework sensitivity stability related measure Extension of the pole-sensitivity stability related measure 3 Implicit State-Space Framework Pole Optimal realization 4 Sensitivity Measure Optimal Conclusion and Perspectives 5 realization Conclusion 4/31
FWL degradation ROCOND’06 — T. Hilaire, P. Chevrel, Origin of the degradation J.P. Clauzel The deterioration induced by the FWL implementation comes Introduction from : A pole sensitivity Quantization of the involved coefficients stability related → parametric errors measure Implicit Roundoff noises in numerical computations State-Space Framework → numerical noises Pole Sensitivity Measure Only the deterioration induced by the quantization of Optimal coefficients is considered here. realization Conclusion 5/31
FWL degradation ROCOND’06 — T. Hilaire, P. Chevrel, Origin of the degradation J.P. Clauzel The deterioration induced by the FWL implementation comes Introduction from : A pole sensitivity Quantization of the involved coefficients stability related → parametric errors measure Implicit Roundoff noises in numerical computations State-Space Framework → numerical noises Pole Sensitivity Measure Only the deterioration induced by the quantization of Optimal coefficients is considered here. realization Conclusion 5/31
Problem setup ROCOND’06 — Let’s consider a discrete plant P T. Hilaire, � X p P. Chevrel, A p X p J.P. Clauzel = k + B p ( R k + Y k ) k +1 P C p X p R k U k = U k Introduction + P k A pole sensitivity and a LTI controller C Y k stability C � X k +1 related measure = AX k + BU k C Implicit Y k = CX k + DU k State-Space Framework Pole The realizations of the form ( T − 1 AT , T − 1 B , CT , D ), with T a Sensitivity Measure non-singular matrix, are all equivalent in infinite precision. Optimal They are no more in finite precision. realization The degradation of the realization depends on the realization. Conclusion 6/31
Problem setup ROCOND’06 — Let’s consider a discrete plant P T. Hilaire, � X p P. Chevrel, A p X p J.P. Clauzel = k + B p ( R k + Y k ) k +1 P C p X p R k U k = U k Introduction + P k A pole sensitivity and a LTI controller C Y k stability C � X k +1 related measure = AX k + BU k C Implicit Y k = CX k + DU k State-Space Framework Pole The realizations of the form ( T − 1 AT , T − 1 B , CT , D ), with T a Sensitivity Measure non-singular matrix, are all equivalent in infinite precision. Optimal They are no more in finite precision. realization The degradation of the realization depends on the realization. Conclusion 6/31
Problem setup ROCOND’06 — � D � C When quantized, the parameters X � T. Hilaire, are changed in P. Chevrel, B A J.P. Clauzel X + ∆ X and the closed-loop system can became unstable. Introduction λ k (¯ � � Let’s denote A ( X )) 1 � k � l the eigenvalues of the A pole sensitivity closed-loop system stability related � ¯ measure A ¯ ¯ X k + ¯ X k +1 = BR k Implicit (1) State-Space C ¯ ¯ = U k X k Framework Pole with Sensitivity Measure � A p + B p DC p � B p C ¯ A � Optimal BC p A realization (2) � B p � Conclusion ¯ ¯ B � C � � � 0 C p 0 7/31
Problem setup ROCOND’06 — � D � C When quantized, the parameters X � T. Hilaire, are changed in P. Chevrel, B A J.P. Clauzel X + ∆ X and the closed-loop system can became unstable. Introduction λ k (¯ � � Let’s denote A ( X )) 1 � k � l the eigenvalues of the A pole sensitivity closed-loop system stability related � ¯ measure A ¯ ¯ X k + ¯ X k +1 = BR k Implicit (1) State-Space C ¯ ¯ = U k X k Framework Pole with Sensitivity Measure � A p + B p DC p � B p C ¯ A � Optimal BC p A realization (2) � B p � Conclusion ¯ ¯ B � C � � � 0 C p 0 7/31
Pole-sensitivity stability related measure Chen, Wu, Li,... (2000-2005) have proposed a pole-sensitivity ROCOND’06 — measure defined by T. Hilaire, P. Chevrel, � λ k (¯ � � 1 − A ( X )) J.P. Clauzel � µ ( X ) � min √ N Ψ k 1 � k � r Introduction with N is the number of non-trivial elements in X (non-zero A pole sensitivity elements in ∆ X ) and Ψ k is the pole sensitivity of the stability related closed-loop with respect to the parameters : measure Implicit 2 � � λ k (¯ � � � State-Space ∂ A ( X )) � � � � Framework Ψ k � ( W X ) i , j � � � ∂ X i , j � Pole � � i , j Sensitivity Measure W X is the weighting matrix associated to the realization matrix Optimal X , defined by realization Conclusion � 0 if X i , j is exactly implemented ( W X ) i , j = 1 if not 8/31
Pole-sensitivity stability related measure Chen, Wu, Li,... (2000-2005) have proposed a pole-sensitivity ROCOND’06 — measure defined by T. Hilaire, P. Chevrel, � λ k (¯ � � 1 − A ( X )) J.P. Clauzel � µ ( X ) � min √ N Ψ k 1 � k � r Introduction with N is the number of non-trivial elements in X (non-zero A pole sensitivity elements in ∆ X ) and Ψ k is the pole sensitivity of the stability related closed-loop with respect to the parameters : measure Implicit 2 � � λ k (¯ � � � State-Space ∂ A ( X )) � � � � Framework Ψ k � ( W X ) i , j � � � ∂ X i , j � Pole � � i , j Sensitivity Measure W X is the weighting matrix associated to the realization matrix Optimal X , defined by realization Conclusion � 0 if X i , j is exactly implemented ( W X ) i , j = 1 if not 8/31
Pole-sensitivity stability related measure Chen, Wu, Li,... (2000-2005) have proposed a pole-sensitivity ROCOND’06 — measure defined by T. Hilaire, P. Chevrel, � λ k (¯ � � 1 − A ( X )) J.P. Clauzel � µ ( X ) � min √ N Ψ k 1 � k � r Introduction with N is the number of non-trivial elements in X (non-zero A pole sensitivity elements in ∆ X ) and Ψ k is the pole sensitivity of the stability related closed-loop with respect to the parameters : measure Implicit 2 � � λ k (¯ � � � State-Space ∂ A ( X )) � � � � Framework Ψ k � ( W X ) i , j � � � ∂ X i , j � Pole � � i , j Sensitivity Measure W X is the weighting matrix associated to the realization matrix Optimal X , defined by realization Conclusion � 0 if X i , j is exactly implemented ( W X ) i , j = 1 if not 8/31
Pole-sensitivity stability related measure ROCOND’06 — T. Hilaire, the measure is such that P. Chevrel, J.P. Clauzel � ∆ X � max � µ ( X ) ⇒ ¯ A ( X + ∆ X ) is stable Introduction A pole sensitivity it considers how close the eigenvalues are to 1 and how stability related sensitive they are w.r.t the controller parameters ; measure this measure is directly linked an estimation of the Implicit State-Space smallest word-length bit needed to guarantee the Framework closed-loop stability Pole Sensitivity Measure the optimal design problem associated consists in finding Optimal an equivalent realization ( T − 1 AT , T − 1 B , CT , D ) that realization maximizes this measure. Conclusion 9/31
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