Examples of Set-Estimation Problems Interval Tools for Set Estimation Returning to Examples Conclusions Interval Analysis for Guaranteed Set Estimation MaGiX@LiX – September 2011 Eric Walter (joint work with L. Jaulin, M. Kieffer et al .) Laboratoire des Signaux et Systèmes CNRS – Supélec – Univ Paris-Sud September 16, 2011 1/54
Examples of Set-Estimation Problems Interval Tools for Set Estimation Returning to Examples Conclusions Outline Examples of Set-Estimation Problems 1 Robotics Robust control Parameter estimation Interval Tools for Set Estimation 2 Computing with intervals Solving systems of nonlinear equations Inverting relations between sets Optimizing nonconvex cost functions Robust tuning Returning to Examples 3 Robotics Robust control Parameter estimation 2/54
Examples of Set-Estimation Problems Robotics Interval Tools for Set Estimation Robust control Returning to Examples Parameter estimation Conclusions Examples of Set-Estimation Problems Robotics Stewart-Gough platform, archetypal parallel robot used, e.g., in flight simulators. Given the (fixed) lengths of the six limbs and geometry, find all possible configurations of mobile plate wrt base. Benchmark problem in computer algebra. 3/54
Examples of Set-Estimation Problems Robotics Interval Tools for Set Estimation Robust control Returning to Examples Parameter estimation Conclusions Examples of Set-Estimation Problems Robust control 2 + y u p p c s c 1 3 2 1 s 2 2 ( 1)( ) p s s p s p 2 3 3 For given values of parameters c 1 and c 2 of PI controller, find � � all values of process parameter vector p 1 p 2 p 3 such that behavior of controlled system is acceptable. Most basic requirement is stability. For given set of possible values for process parameter vector, find a tuning of controller parameters that guarantees acceptable behavior (or prove that none exists). 4/54
Examples of Set-Estimation Problems Robotics Interval Tools for Set Estimation Robust control Returning to Examples Parameter estimation Conclusions Examples of Set-Estimation Problems Guaranteed parameter estimation ( ) y t System m ( , ) y t Model p M p ( ) Find all values of parameter vector p such that error between system output and model output belongs to some acceptable set. Find all values of p that are optimal in some statistical sense. 5/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Origins of Interval Computations Used by Archimedes, 3rd century BC to enclose � high-school physicists to assess errors... Systematic use in numerical analysis and on computers often attributed to Ramon Moore (circa 1960), but many precursors, including M. Warmus (1956) and T. Sunaga (1958). Limited impact outside inner circle until beginning of the 90s for various reasons, including implementation issues. Things are improving as we shall see. 6/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Interval Arithmetics Basic arithmetic operations easily extend to intervals: [ x ] ◦ [ y ] = { x ◦ y | x ∈ [ x ] and y ∈ [ y ] } For instance � � [ x ]+[ y ] = x + y , x + y , � � [ x ] − [ y ] = x − y , x − y . Note that [ x ] − [ x ] is not equal to zer o! Try to avoid multi-occurences of variables... results computed using bounds of interval operands (intervals described by pairs of real numbers, just as complex numbers), division tricky when zero belongs to denominator interval. 7/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Inclusion functions Inclusion function [ f ]( · ) of f ( · ) satisfies ∀ [ x ] ⊂ R , f ([ x ]) ⊂ [ f ]([ x ]) . It is minimal if ⊂ can be replaced by = , convergent if width ([ x ]) → 0 ⇒ width ([ f ]([ x ])) → 0 Easy to build for monotone functions, e.g., exp ([ x ]) = [ exp ( x ) , exp ( x )] . (Simple) algorithms available for sin ( · ) , cos ( · ) , etc. Inclusion functions also available for solutions of nonlinear ODEs, see Berz – Makino talk. 8/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Examples of Natural Inclusion Functions Natural inclusion functions: replace each variable and operator by its interval counterpart in formal expression. Consider these four formal expressions of the same function f 3 ( x ) = x 2 + x , f 1 ( x ) = x ( x + 1 ) , 2 ) 2 − 1 f 4 ( x ) = ( x + 1 f 2 ( x ) = x × x + x , 4 . Evaluate their natural inclusion functions for [ x ] = [ − 1 , 1 ] . 9/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Examples of Natural Inclusion Functions Only [ f 4 ] is minimal ( x appears only once). 10/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Another Useful Type of Inclusion Function If f is di ff erentiable over [ x ] , mean-value theorem states that ∀ x ∈ [ x ] , ∃ � ∈ [ x ] such that f ( x ) = f ( m )+ f � ( � ) · ( x − m ) , with m the center of [ x ] . Then f ( x ) ∈ f ( m )+ f � ([ x ]) · ( x − m ) and f � � � f ([ x ]) ⊆ f ( m )+ ([ x ]) · ([ x ] − m ) . Hence the centred form f � � � [ f ] c ([ x ]) = f ( m )+ ([ x ]) · ([ x ] − m ) . 11/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Which Inclusion Function To Use? Compare the natural and centred inclusion functions for f ( x ) = x 2 exp ( x ) − x exp � x 2 � . Best inclusion function depends on width of interval argument: [ x ] f ([ x ]) [ f ]([ x ]) [ f ] c ([ x ]) [ 0 . 5 , 1 . 5 ] [ − 4 . 148 , 0 ] [ − 13 . 82 , 9 . 44 ] [ − 25 . 07 , 25 . 07 ] [ 0 . 9 , 1 . 1 ] [ − 0 . 05380 , 0 ] [ − 1 . 697 , 1 . 612 ] [ − 0 . 5050 , 0 . 5050 ] [ 0 . 99 , 1 . 01 ] [ − 0 . 0004192 , 0 ] [ − 0 . 1636 , 0 . 1628 ] [ − 0 . 0047 , 0 . 0047 ] Intersecting results provided by several inclusion functions may provide a more accurate result than any of them separately. 12/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Computing with Interval Vectors and Functions An interval vector (or box) is a Cartesian product of scalar intervals [ x ] = [ x 1 ] ×··· [ x n ] . Interval computation extends easily to boxes, as well as notion of inclusion function. 13/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Solving Systems of Nonlinear Equations System writen as f ( x ) = 0 . We assume that there are as many equations are there are unknowns (dim f ( x ) = dim x = n ), f is continuously di ff erentiable. We want all solutions in a given box [ x ] 0 . The approach is an interval variant of the Newton method. 14/54
Computing with intervals Examples of Set-Estimation Problems Solving systems of nonlinear equations Interval Tools for Set Estimation Inverting relations between sets Returning to Examples Optimizing nonconvex cost functions Conclusions Robust tuning Interval Newton Method Mean-value theorem implies that ∀ x ∈ [ x ] , ∃ � ∈ [ x ] such that f ( x ) = f ( m )+ J f ( � )( x − m ) . Now we want f ( x ) = 0 , so f ( m )+ J f ( � )( x − m ) = 0 . with J f the Jacobian matrix of f , assumed invertible for the sake of simplicity. Thus x = m − J − 1 f ( � ) f ( m ) . Now, since the value of � is not known, we can only write x ∈ m − J − 1 ([ x ]) f ( m ) , f or rather � � m − J − 1 x ∈ ([ x ]) f ( m ) ∩ [ x ] . f 15/54
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