Towards Guaranteed Accuracy Computations in Control Masaaki Kanno Niigata University Asian Symposium on Computer Mathematics 2012 Organized Session: On The Latest Progress In Verified Computation 27 O CTOBER 2012 Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 1 / 21
Outline Control Theory 1 Design Approach 2 Guaranteed Accuracy Polynomial Spectral Factorization 3 Concluding Remarks 4 Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 2 / 21
Outline Control Theory 1 Design Approach 2 Guaranteed Accuracy Polynomial Spectral Factorization 3 Concluding Remarks 4 Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 3 / 21
Feedback Control Systems Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 4 / 21
Feedback Loop d y ❄ e u r ✲ ❡ ✲ ✲ ❡ ✲ ✲ K P r − ✻ P : Plant — System to be controlled dynamical system K : Controller — Control strategy Aims Stabilization Disturbance attenuation Robustness — strong against uncertainty Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 5 / 21
(Post-)Modern Control Theory Classical Control Qualitative Graphical approach (Simple) Algebraic computation (Post-)Modern Control For superior design.... Mathematically oriented approach Extensive computation — get along with computers Mathematical modelling Mathematical formulation Quantitatively Optimization problems Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 6 / 21
Outline Control Theory 1 Design Approach 2 Guaranteed Accuracy Polynomial Spectral Factorization 3 Concluding Remarks 4 Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 7 / 21
Sketch (1) Dynamical system description High-order (linear) differential equation E.g., ¨ y ( t ) + 3 ˙ y ( t ) + 5 y ( t ) = ˙ u ( t ) + 2 u ( t ) Laplace Transform Set of 1st order differential eqns Transfer function State-space representation introduction of state x ( t ) Y ( s ) = L [ y ( t )] � ˙ x ( t ) = Ax ( t ) + Bu ( t ) U ( s ) = L [ u ( t )] P : y ( t ) = Cx ( t ) P ( s ) = (rational function in s ) � − 3 � � 1 � = Y ( s ) − 5 ˙ x ( t ) = x ( t ) + u ( t ) 1 0 0 U ( s ) � �� � ���� s + 2 A B = s 2 + 3 s + 5 � � y ( t ) = 1 2 x ( t ) � �� � C Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 8 / 21
Sketch (2) Laplace Transformation Set of 1st order differential eqns Polynomial Algebraic Riccati equation 1 1 Spectral Factorization XA + A T X f ( s ) � − XBB T X + C T C = 0 Y ( s ) Y ( − s ) + U ( s ) U ( − s ) YA T + AY = g ( s ) g ( − s ) − YC T CY + BB T = 0 Calculating the Controller 2 Calculating the Controller 2 Set of linear equations Straightforward matrix computation x K ( t ) = ( A − YC T C − BB T X ) x K ( t ) ˙ ! In either approach, + YC T y ( t ) K : Step 1 is the harder. u ( t ) = − B T Xx K ( t ) Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 9 / 21
Normalized LQG Control Problem Problem Formulation LQG = Linear Quadratic Gaussian d 1 y 1 ✲ ❡ ✲ P � � ✻ min � T zw ( P , K ) � 2 K stabilizing y 2 d 2 ❄ ✛ ✛ K ❡ Given P , find a controller K that minimizes the H 2 -norm of the transfer function matrix T zw from w = ( d 1 d 2 ) T to z = ( y 1 y 2 ) T . H 2 -norm � ∞ � 1 � 1 � � 2 2 � tr { G ∗ ( i ω ) G ( i ω ) } d ω � G ( s ) � 2 π −∞ � � � 2 � G ( s ) 2 : Energy of the system output to an impulse input signal Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 10 / 21
Outline Control Theory 1 Design Approach 2 Guaranteed Accuracy Polynomial Spectral Factorization 3 Concluding Remarks 4 Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 11 / 21
What is Polynomial Spectral Factorization? Given : f ( s ) = f ( − s ) = − s 6 + 9 s 4 − 4 s 2 + 36 ; Self-reciprocal polynomial Task : Decompose f ( s ) as a product of a stable polynomial and an anti-stable polynomial (‘ mirror image ’) � s 3 + 5 s 2 + 8 s + 6 � � − s 3 + 5 s 2 − 8 s + 6 � f ( s ) = � �� � � �� � g ( s ) g ( − s ) Self-reciprocal — stable and unstable roots symmetrically ✻ Im Stable: all the roots in • • the left half plane Re ✲ Self-reciprocal : f ( s ) • • Stable : g ( s ) — spectral factor • • Mirror image : g ( − s ) Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 12 / 21
What is Polynomial Spectral Factorization? Given : f ( s ) = f ( − s ) = − s 6 + 9 s 4 − 4 s 2 + 36 ; Self-reciprocal polynomial Task : Decompose f ( s ) as a product of a stable polynomial and an anti-stable polynomial (‘ mirror image ’) � s 3 + 5 s 2 + 8 s + 6 � � − s 3 + 5 s 2 − 8 s + 6 � f ( s ) = � �� � � �� � g ( s ) g ( − s ) Self-reciprocal — stable and unstable roots symmetrically ✻ Im Stable: all the roots in • • the left half plane Re ✲ Self-reciprocal : f ( s ) • • Stable : g ( s ) — spectral factor • • Mirror image : g ( − s ) Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 12 / 21
What is Polynomial Spectral Factorization? Given : f ( s ) = f ( − s ) = − s 6 + 9 s 4 − 4 s 2 + 36 ; Self-reciprocal polynomial Task : Decompose f ( s ) as a product of a stable polynomial and an anti-stable polynomial (‘ mirror image ’) � s 3 + 5 s 2 + 8 s + 6 � � − s 3 + 5 s 2 − 8 s + 6 � f ( s ) = � �� � � �� � g ( s ) g ( − s ) Self-reciprocal — stable and unstable roots symmetrically ✻ Im Stable: all the roots in • • the left half plane Re ✲ Self-reciprocal : f ( s ) • • Stable : g ( s ) — spectral factor • • Mirror image : g ( − s ) Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 12 / 21
‘Full’ Guaranteed Approach By Means of Verified Polynomial Root Computation Each Root in the left half plane is found as an interval on the real axis / a box in the complex plane. Express the spectral factor as a product of linear factors and expand it to get bounds for coefficients g ( s ) = ( s − p 1 )( s − p 2 )( s − p 3 ) = s 3 − ( p 1 + p 2 + p 3 ) s 2 + ( p 1 p 2 + p 2 p 3 + p 3 p 1 ) s − p 1 p 2 p 3 ✻ Im • • Re ✲ • • • • Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 13 / 21
‘Full’ Guaranteed Approach By Means of Verified Polynomial Root Computation Each Root in the left half plane is found as an interval on the real axis / a box in the complex plane. Express the spectral factor as a product of linear factors and expand it to get bounds for coefficients g ( s ) = ( s − p 1 )( s − p 2 )( s − p 3 ) = s 3 − ( p 1 + p 2 + p 3 ) s 2 + ( p 1 p 2 + p 2 p 3 + p 3 p 1 ) s − p 1 p 2 p 3 ✻ Im • • Re ✲ • • • • Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 13 / 21
‘Full’ Guaranteed Approach By Means of Verified Polynomial Root Computation Each Root in the left half plane is found as an interval on the real axis / a box in the complex plane. Express the spectral factor as a product of linear factors and expand it to get bounds for coefficients g ( s ) = ( s − p 1 )( s − p 2 )( s − p 3 ) = s 3 − ( p 1 + p 2 + p 3 ) s 2 + ( p 1 p 2 + p 2 p 3 + p 3 p 1 ) s − p 1 p 2 p 3 ✻ Im • • Re ✲ • • • • Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 13 / 21
Hybrid Approach ‘Full’ Guaranteed Approach — always works Eventually want to get bounds for coefficients of the spectral factor To get tighter bounds for coefficients, the Krawczyk method can be employed. Suggested Approach Compute coefficients of the spectral factor 1 using an ordinary ( unverified ) numerical method Give ( heuristically ) bounds for coefficients, 2 and check whether a solution is included in the bounds Use ‘Full’ Guaranteed Approach as a backup Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 14 / 21
Polynomial Spectral Factorization Problem Formulation Given : an even polynomial in s (polynomial in s 2 ) f ( s ) = ( − 1 ) n s 2 n + a 2 n − 2 s 2 n − 2 + a 2 n − 4 s 2 n − 4 + · · · + a 0 Task : Find a polynomial g ( s ) = s n + b n − 1 s n − 1 + b n − 2 s n − 2 + · · · + b 0 f ( s ) = ( − 1 ) n g ( s ) g ( − s ) such that and g ( s ) has roots in the open left half plane only. By comparing the coefficients of the both sides of f ( s ) = ( − 1 ) n g ( s ) g ( − s ) , a set of algebraic equations in b j is obtained. Krawczyk method easy to apply Masaaki Kanno ( Niigata University) Towards Guaranteed Accuracy Computations in Control ASCM 2012 15 / 21
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