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Thibault Thibault HILAIRE HILAIRE thibault.hilaire@irisa.fr thibault.hilaire@irisa.fr Bit Accurate Roundoff Bit Accurate Roundoff Noise Analysis of Noise Analysis of Fixed-Point Linear Fixed-Point Linear Controllers Controllers IEEE Multi-Conference on Systems and Control San Antonio, Texas CAIRN project 4 September 2008 IRISA/INRIA France

Introduction Context Context Target filter/controller 0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 H ( z ) = 0 . 004708 z 6 − 0 . 0251 1 0 z 6 − 5 . 653 . 0251 z 5 + 0 . 05844 z 4 − 0 . 07608 z 5 . 653 z 5 + 13 . 38 z 4 − 16 . 98 z 3 + 8 z 3 + 0 . 05844 z 2 − 0 . 0251 z + 0 . + 12 . 18 z 2 − 4 . 679 z + 0 . 7526 X k +1 = AX k + BU k Y k = CX k + DU k Implementation of Linear Time Invariant controllers/filters Finite Word Length context (fixed-point) T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 2/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Introduction Context Context Target filter/controller 0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 H ( z ) = 0 . 004708 z 6 − 0 . 0251 1 0 z 6 − 5 . 653 . 0251 z 5 + 0 . 05844 z 4 − 0 . 07608 z 5 . 653 z 5 + 13 . 38 z 4 − 16 . 98 z 3 + 8 z 3 + 0 . 05844 z 2 − 0 . 0251 z + 0 . + 12 . 18 z 2 − 4 . 679 z + 0 . 7526 X k +1 = AX k + BU k Y k = CX k + DU k Implementation of Linear Time Invariant controllers/filters Finite Word Length context (fixed-point) T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 2/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Introduction Context Context Target filter/controller 0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 H ( z ) = 0 . 004708 z 6 − 0 . 0251 1 0 z 6 − 5 . 653 . 0251 z 5 + 0 . 05844 z 4 − 0 . 07608 z 5 . 653 z 5 + 13 . 38 z 4 − 16 . 98 z 3 + 8 z 3 + 0 . 05844 z 2 − 0 . 0251 z + 0 . + 12 . 18 z 2 − 4 . 679 z + 0 . 7526 X k +1 = AX k + BU k Y k = CX k + DU k Implementation of Linear Time Invariant controllers/filters Finite Word Length context (fixed-point) Motivation Analysis (accurately) the roundoff noise errors in the implementation Compare various realizations and find an optimal one T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 2/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Introduction Context Context Target filter/controller 0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 H ( z ) = 0 . 004708 z 6 − 0 . 0251 1 0 z 6 − 5 . 653 . 0251 z 5 + 0 . 05844 z 4 − 0 . 07608 z 5 . 653 z 5 + 13 . 38 z 4 − 16 . 98 z 3 + 8 z 3 + 0 . 05844 z 2 − 0 . 0251 z + 0 . + 12 . 18 z 2 − 4 . 679 z + 0 . 7526 X k +1 = AX k + BU k Y k = CX k + DU k Implementation of Linear Time Invariant controllers/filters Finite Word Length context (fixed-point) The roundoff will depend on the algorithmic relation to compute the output(s) from the input(s) the way the computations are implemented (wordlength, roundoff, etc.) T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 2/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Introduction Outline Outline 1 Implicit state-space framework 2 Roundoff noise analysis 3 Fixed-point implementation schemes 4 Optimal design 5 Conclusion T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 3/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Implicit Implicit ✶ ✶ State-Space State-Space Framework Framework

Implicit State-Space Framework The need of a unifying framework The need of a unifying framework Various implementation forms have to be taken into consideration: shift-realizations δ -realizations observer-state-feedback direct form I or II cascade or parallel realizations etc... T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 5/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Implicit State-Space Framework Specialized Implicit Form (SIF) Specialized Implicit Form (SIF) Implicit specialized state-space form         J 0 0 T k +1 0 M N T k  = − K I 0 X k +1 0 P Q X k        − L 0 I Y k 0 R S U k T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 6/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Implicit State-Space Framework Specialized Implicit Form (SIF) Specialized Implicit Form (SIF) Implicit specialized state-space form         J 0 0 T k +1 0 M N T k  = − K I 0 X k +1 0 P Q X k        − L 0 I Y k 0 R S U k It corresponds to: 1 J . T k +1 = M . X k + N . U k 2 X k +1 = K . T k +1 + P . X k + Q . U k 3 Y k = L . T k +1 + R . X k + S . U k Intermediate variables computation T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 6/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Implicit State-Space Framework Specialized Implicit Form (SIF) Specialized Implicit Form (SIF) Implicit specialized state-space form         J 0 0 T k +1 0 M N T k  = − K I 0 X k +1 0 P Q X k        − L 0 I Y k 0 R S U k It corresponds to: 1 J . T k +1 = M . X k + N . U k 2 X k +1 = K . T k +1 + P . X k + Q . U k 3 Y k = L . T k +1 + R . X k + S . U k State-space computation T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 6/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Implicit State-Space Framework Specialized Implicit Form (SIF) Specialized Implicit Form (SIF) Implicit specialized state-space form         J 0 0 T k +1 0 M N T k  = − K I 0 X k +1 0 P Q X k        − L 0 I Y k 0 R S U k It corresponds to: 1 J . T k +1 = M . X k + N . U k 2 X k +1 = K . T k +1 + P . X k + Q . U k 3 Y k = L . T k +1 + R . X k + S . U k Output(s) computation T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 6/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Implicit State-Space Framework Specialized Implicit Form (SIF) Specialized Implicit Form (SIF) Implicit specialized state-space form         J 0 0 T k +1 0 M N T k  = − K I 0 X k +1 0 P Q X k        − L 0 I Y k 0 R S U k It is equivalent to the system H : z �→ C Z ( zI n − A Z ) B Z + D Z with � A Z � � K � � P � B Z Q J − 1 � � � M N + C Z D Z L R S T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 6/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Implicit State-Space Framework Intermediate variables Intermediate variables The intermediate variables introduced allow to make explicit all the computations done show the order of the computations express a larger parametrization Implicit realization The intermediate variables computation is expressed by J . T k +1 = M . X k + N . U k with J lower triangular with 1 on diagonal, so no need to compute J − 1 an intermediate variable may be computed from another one previously computed (in the same stage) ⇒ can express realizations like Y k = M 1 . M 2 ... M i U k T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 7/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Implicit State-Space Framework Intermediate variables Intermediate variables The intermediate variables introduced allow to make explicit all the computations done show the order of the computations express a larger parametrization Implicit realization The intermediate variables computation is expressed by J . T k +1 = M . X k + N . U k with J lower triangular with 1 on diagonal, so no need to compute J − 1 an intermediate variable may be computed from another one previously computed (in the same stage) ⇒ can express realizations like Y k = M 1 . M 2 ... M i U k T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 7/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers

Implicit State-Space Framework Example Example A realization with the δ -operator is described by : � δ X k = A δ X k + B δ U k δ � q − 1 ∆ Y k = C δ X k + D δ U k It is computed with  T = A δ X k + B δ U k  = X k + ∆ T X k +1 Y k = C δ X k + D δ U k  and it corresponds to the following implicit state-space :         0 0 0 I T k +1 A δ B δ T k  = − ∆ I I 0 X k +1 0 I 0 X k        0 0 0 I Y k C δ D δ U k T. Hilaire Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers 8/34 Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers