Filter Context Realizations Evaluation Approximation Conclusion Current state on filter approximation and evaluation Thibault Hilaire (thibault.hilaire@lip6.fr) Kick-off ANR MetaLibm — January 22, 2014 Filter approximation and evaluation T. Hilaire 1/22
Filter Context Realizations Evaluation Approximation Conclusion We consider here linear filters • bricks for signal processing algorithms • linear controllers are linear filters (used with feedback) Signals • Continous-time signal: s ( t ) , t ∈ R • Discrete-time signal: s ′ ( k ) , k ∈ Z s ′ ( k ) � s ( kT e ) , T e sampling period Filter approximation and evaluation T. Hilaire 2/22
Filter Context Realizations Evaluation Approximation Conclusion We consider here linear filters • bricks for signal processing algorithms • linear controllers are linear filters (used with feedback) Signals • Continous-time signal: s ( t ) , t ∈ R • Discrete-time signal: s ′ ( k ) , k ∈ Z s ′ ( k ) � s ( kT e ) , T e sampling period Filter A filter is defined by u ( k ) y ( k ) • its impulse response H • its transfer function • its frequency response Filter approximation and evaluation T. Hilaire 2/22
Filter Context Realizations Evaluation Approximation Conclusion Transfer function H ( z ) = Y ( z ) U ( z ) where X ( z ) and Y ( z ) are the Z -transform of u ( k ) and y ( k ) ( H ( z ) is the Z -transform of the impulse response h ( k ) ) Filter approximation and evaluation T. Hilaire 3/22
Filter Context Realizations Evaluation Approximation Conclusion Transfer function H ( z ) = Y ( z ) U ( z ) where X ( z ) and Y ( z ) are the Z -transform of u ( k ) and y ( k ) ( H ( z ) is the Z -transform of the impulse response h ( k ) ) Z -transform It is the discrete equivalent of the Laplace -transform: Z [ x ( k )] : D cv → C z X ( z ) �→ with ∞ � x ( k ) z − k X ( z ) � k = 0 Filter approximation and evaluation T. Hilaire 3/22
Filter Context Realizations Evaluation Approximation Conclusion Frequency response Frequency response: H ( e j ω ) • for z = e j ω , ω ∈ [ 0 , 2 π ] • for z = e j 2 π f Fe with 0 � f � Fe /9+,-:(%&;%< !# !" $%&'()*+,-.+/0 # 1 H ( z ) = " z − 0 . 8 ! # � H ( e j Ω ) � � ! !" " � � H ( e j Ω ) � arg ! 5# 67%8,-.+,&0 ! 4" ! !3# ! !2" ! 1 ! ! " ! !" !" !" !" =;,>*,'?@--.;%+A8,?0 Filter approximation and evaluation T. Hilaire 4/22
Filter Context Realizations Evaluation Approximation Conclusion • LTI: Linear system with known and constant coefficients • FIR (Finite Impulse Response) : n n � � b i z − i , H ( z ) = ⇒ y ( k ) = b i u ( k − i ) i = 0 i = 0 • IIR (Infinite Impulse Response) : � n n n i = 0 b i z − i � � H ( z ) = ⇒ y ( k ) = b i u ( k − i ) − a i y ( k − i ) i = 1 a i z − i , 1 + � n i = 0 i = 1 Filter approximation and evaluation T. Hilaire 5/22
Filter Context Realizations Evaluation Approximation Conclusion • LTI: Linear system with known and constant coefficients • FIR (Finite Impulse Response) : n n � � b i z − i , H ( z ) = ⇒ y ( k ) = b i u ( k − i ) i = 0 i = 0 • IIR (Infinite Impulse Response) : � n n n i = 0 b i z − i � � H ( z ) = ⇒ y ( k ) = b i u ( k − i ) − a i y ( k − i ) i = 1 a i z − i , 1 + � n i = 0 i = 1 • parametrized LTI: Linear system, where the coefficients are constant, but dépends on extra parameters (unknown at compile-time) Filter approximation and evaluation T. Hilaire 5/22
Filter Context Realizations Evaluation Approximation Conclusion • LTI: Linear system with known and constant coefficients • FIR (Finite Impulse Response) : n n � � b i z − i , H ( z ) = ⇒ y ( k ) = b i u ( k − i ) i = 0 i = 0 • IIR (Infinite Impulse Response) : � n n n i = 0 b i z − i � � H ( z ) = ⇒ y ( k ) = b i u ( k − i ) − a i y ( k − i ) i = 1 a i z − i , 1 + � n i = 0 i = 1 • parametrized LTI: Linear system, where the coefficients are constant, but dépends on extra parameters (unknown at compile-time) • LPV : Linear Parameter Varying (coefficients depends on time) Filter approximation and evaluation T. Hilaire 5/22
Filter Context Realizations Evaluation Approximation Conclusion • LTI: Linear system with known and constant coefficients • FIR (Finite Impulse Response) : n n � � b i z − i , H ( z ) = ⇒ y ( k ) = b i u ( k − i ) i = 0 i = 0 • IIR (Infinite Impulse Response) : � n n n i = 0 b i z − i � � H ( z ) = ⇒ y ( k ) = b i u ( k − i ) − a i y ( k − i ) i = 1 a i z − i , 1 + � n i = 0 i = 1 • parametrized LTI: Linear system, where the coefficients are constant, but dépends on extra parameters (unknown at compile-time) • LPV : Linear Parameter Varying (coefficients depends on time) → Can be with multiple inputs and multiple outputs (MIMO controllers). Filter approximation and evaluation T. Hilaire 5/22
Filter Context Realizations Evaluation Approximation Conclusion Given • a filter (transfer function or frequency specification) • some requirement on the acceptable performance / error • a specific hardware generate code with guarantee (on the error/performance) Filter approximation and evaluation T. Hilaire 6/22
Filter Context Realizations Evaluation Approximation Conclusion Given • a filter (transfer function or frequency specification) • some requirement on the acceptable performance / error • a specific hardware generate code with guarantee (on the error/performance) Several metrics are used to evaluate the distance to original real filter • maximal output error (with/without knowledge on the input) • transfer function error • poles and/or zeros error (distance to instability) • etc. Filter approximation and evaluation T. Hilaire 6/22
Filter Context Realizations Evaluation Approximation Conclusion Equivalent realizations For a given LTI controller, it exist various equivalent realizations • Direct Form I (2 n + 1 coefs) u ( k ) u ( k − 1) u ( k − n ) q − 1 q − 1 q − 1 q − 1 b 0 b 1 b 2 b i b n � n i = 0 b i z − i H ( z ) = 1 + � n i = 1 a i z − i + y ( k ) n n � � y ( k ) = b i u ( k − i ) − a i y ( k − i ) − a n − a i − a 2 − a 1 i = 0 i = 1 q − 1 q − 1 q − 1 q − 1 y ( k − n ) y ( k − 1) Filter approximation and evaluation T. Hilaire 7/22
Filter Context Realizations Evaluation Approximation Conclusion Equivalent realizations For a given LTI controller, it exist various equivalent realizations • Direct Form I (2 n + 1 coefs) • Direct Form II (transposed or not) (2 n + 1 coefs) e ( k ) y ( k ) u ( k ) + + b 0 q − 1 e ( k − 1) + + − a 1 b 1 q − 1 + + b 2 − a 2 q − 1 + + − a i b i q − 1 − a n b n e ( k − n ) Filter approximation and evaluation T. Hilaire 7/22
Filter Context Realizations Evaluation Approximation Conclusion Equivalent realizations For a given LTI controller, it exist various equivalent realizations • Direct Form I (2 n + 1 coefs) • Direct Form II (transposed or not) (2 n + 1 coefs) • State-space (depend on the basis) ( ( n + 1 ) 2 coefs) � X ( k + 1 ) = AX ( k ) + Bu ( k ) y ( k ) = CX ( k ) + Du ( k ) A X ( k ) → T . X ( k ) U ( k ) Y ( k ) q − 1 C B + + X ( k + 1) X ( k ) D Filter approximation and evaluation T. Hilaire 7/22
Filter Context Realizations Evaluation Approximation Conclusion Equivalent realizations For a given LTI controller, it exist various equivalent realizations • Direct Form I (2 n + 1 coefs) • Direct Form II (transposed or not) (2 n + 1 coefs) • State-space (depend on the basis) ( ( n + 1 ) 2 coefs) δ � q − 1 • δ -operator ∆ A δ U ( k ) Y ( k ) B δ δ − 1 C δ + + ∆ + q − 1 X ( k + 1) X ( k ) δ − 1 � δ [ X ( k )] D δ = A δ X ( k ) + B δ u ( k ) y ( k ) = C δ X ( k ) + D δ u ( k ) Filter approximation and evaluation T. Hilaire 7/22
Filter Context Realizations Evaluation Approximation Conclusion Equivalent realizations For a given LTI controller, it exist various equivalent realizations • Direct Form I (2 n + 1 coefs) • Direct Form II (transposed or not) (2 n + 1 coefs) • State-space (depend on the basis) ( ( n + 1 ) 2 coefs) δ � q − 1 • δ -operator ∆ • Cascad and/or parallel decompositions H 1 H 2 + H 1 H 2 H i H m H i H H m H Filter approximation and evaluation T. Hilaire 7/22
Filter Context Realizations Evaluation Approximation Conclusion Equivalent realizations For a given LTI controller, it exist various equivalent realizations • Direct Form I (2 n + 1 coefs) • Direct Form II (transposed or not) (2 n + 1 coefs) • State-space (depend on the basis) ( ( n + 1 ) 2 coefs) δ � q − 1 • δ -operator ∆ • Cascad and/or parallel decompositions • ρ Direct Form II transposed (5 n + 1 ou 4 n + 1 coefs) ρ DFIIt u ( k ) β n β i β 1 β 0 β n − 1 + z − 1 ∆ i + ρ − 1 + ρ − 1 + ρ − 1 + ρ − 1 + n i +1 i 1 γ i ρ − 1 i α n α n − 1 α i α 1 y ( k ) Filter approximation and evaluation T. Hilaire 7/22
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