Singular Value Analyi of Linear- Quadratic Sytem! Robert Stengel! Optimal Control and Etimation MAE 546! Princeton Univerity, 2017!! Multivariable Nyquit Stability Criterion!! Matrix Norm and Singular Value Analyi!! Frequency domain meaure of robutne!! Stability Margin of Multivariable Linear-Quadratic Regulator Copyright 2017 by Robert Stengel. All right reerved. For educational ue only. http://www.princeton.edu/~tengel/mae546.html http://www.princeton.edu/~tengel/optconet.html 1 Scalar Tranfer Function and Return Difference Function!! Unit feedback control law Block diagram algebra!y = A( ) #!y C ( )!y( ) & = A( )!y C ( ) = A( ) 1+ A( ) = A( ) 1+ A( ) # 1+ A &!y!y!y C # & 1 A( ) : Tranfer Function! 1+ A # : Return Difference Function 2
= kn( )! OL ( ) A Relationhip Between SISO Open- and Cloed- Loop Characteritic Polynomial!y!y C = kn( )! OL ( ) 1+ kn( )! OL ( ) kn( ) =! OL ( ) + kn( ) # = # =! OL kn( ) 1+ kn( )! OL ( ) kn! CL #!! Cloed-loop polynomial i open-loop polynomial multiplied by return difference function! CL ( ) =! OL ( ) 1+ A( ) # 3 Return Difference Function Matrix for the Multivariable LQ Regulator Open-loop ytem!x() = F!x() + G!u() ( I F)!x() = G!u()!x() = I F 1 G!u() Linear-quadratic feedback control law!u( ) = R 1 G T P!x( )! C!x( ) = C( I F) 1 G!u()! A!u() 4
Multivariable LQ Regulator Portrayed a a Unit-Feedback Sytem 5 Broken-Loop Analyi of Unit-Feedback Repreentation of LQ Regulator Cut the loop a hown Analyze ignal flow from! ( ) to! ( )!() =!u C () # A( )!() & ' I m + A( ) & '!() =!u C ()!u() = A( )#() = A #1!() = I m + A( ) & '!u C () I m + A( ) & '!1 u C () Analogy to SISO cloed-loop tranfer function 6
Broken-Loop Analyi of Unit-Feedback Repreentation of LQ Regulator Cut the loop a hown Analyze ignal flow from! u( ) to! u( ) = C( I n! F)!1 G u C () + u() A!u() = A( )#() = A! # I m + A( ) &u() = A( )&u C () [ ]!1!&u() = # I m + A( ) A( )&u C () 7 Cloed-Loop Tranfer Function Matrix i Commutative!u() = A I m + A( ) # &!1 u C ()!1!u() = # I m + A( ) & A( )u C () 2 nd -order example A[ I + A]!1 = [ I + A]!1 A # a 1 a 2 a 3 a 4 = # ' &' # a 1 +1 a 2 a 3 a 4 +1 ' &'!1 = # a 1 +1 a 2 a 3 a 4 +1 ( a 1 a 4! a 2 a 3 + a 1 ) 1 a 3 a 1 a 4! a 2 a 3 + a 4 ' ' & ' &'!1 # a 1 a 2 a 3 a 4 det( I 2 + A) ' &' 8
Relationhip Between Multi-Input/Multi- Output (MIMO) Open- and Cloed-Loop Characteritic Polynomial I m + A( ) = I m + C( I n! F)!1 G = I m + CAdj ( I n! F)G OL! OL G I m + CAdj I F n! OL ( ) =! OL ( ) I m + A( ) =! CL ( ) = 0 Cloed-loop polynomial i open-loop polynomial multiplied by determinant of return difference function matrix 9 Multivariable Nyquit Stability Criterion! 10
Ratio of Cloed- to Open-Loop Characteritic Polynomial Teted in Nyquit Stability Criterion! CL! OL = 1+ A( ) Scalar Control # = 1+ CAdj( I & F)G n ' ( #! OL ( ) = a + jb( ) Scalar! CL! OL Multivariate Control G = I m + A ( ) = I + CAdj I F n m! OL ( ) = a( ) + jb( ) Scalar 11 Multivariable Nyquit Stability Criterion! CL! OL = I + A( )! a( ) + jb( ) Scalar m Same tability criteria for encirclement of 1 point apply for calar and vector control Eigenvalue Plot, D Contour Nyquit Plot Nyquit Plot, hifted origin 12
Limit of Multivariable Nyquit Stability Criterion! CL! OL = I m + A( )! a( ) + jb( ) Scalar! Multivariable Nyquit Stability Criterion! Indicate tability of the nominal ytem! In the I + A() plane, Nyquit plot depict the ratio of cloed-to-open-loop characteritic polynomial! However, determinant i not a good indicator for the ize of a matrix! Little can be aid about robutne! Therefore, analogie to gain and phae margin are not readily identified 13 Determinant i Not a Reliable Meaure of Matrix Size! A 1 = #! A 2 = #! A 3 = # 1 0 0 2 1 100 0 2 1 100 0.02 2 &; A 1 = 2 &; A 2 = 2 &; A 3 = 0! Qualitatively,! A 1 and A 2 have the ame determinant! A 2 and A 3 are about the ame ize 14
Matrix Norm and Singular Value Analyi! 15 Vector Norm Euclidean norm x = ( x T x) 1/2 Weighted Euclidean norm Dx = ( x T D T Dx) 1/2 For fixed value of x, Dx provide a meaure of the ize of D 16
Spectral Norm (or Matrix Norm) Spectral norm ha more than one ize D = max x =1 Dx i real-valued dim( x) = dim( Dx) = n!1; dim( D) = n! n Alo called Induced Euclidean norm If D and x are complex 1/2 1/2 x = x H x Dx = x H D H Dx x H! complex conjugate tranpoe of x = Hermitian tranpoe of x 17 Spectral Norm (or Matrix Norm) Spectral norm of D D = max x =1 Dx D T D or D H D ha n eigenvalue Eigenvalue are all real, a D T D i ymmetric and D H D i Hermitian Square root of eigenvalue are called ingular value 18
Singular Value of D Singular value of D! i ( D) = i ( D T D), i = 1,n Maximum ingular value of D! max ( D)!! ( D)! D = max x =1 Minimum ingular value of D Dx! min ( D)!! ( D) = 1 D 1 = min Dx x =1 19 Comparion of Determinant and Singular Value! A 1 = #! A 2 = #! A 3 = # 1 0 0 2 1 100 0 2 1 100 0.02 2 &; A 1 = 2 &; A 2 = 2 &; A 3 = 0 A 1 :! A 1 A 2 :! A 2 A 3 :! A 3!!! Singular value provide a better portrayal of matrix ize, but... Size i multi-dimenional Singular value decribe magnitude along axe of a multidimenional ellipoid e.g., x 2! + y2 2! = 1 2 = 2;! ( A 1 ) = 1 = 100.025;! ( A 2 ) = 0.02 = 100.025;! ( A 3 ) = 0 20
Stability Margin of Multivariable LQ Regulator! 21 Bode Gain Criterion and the Cloed-Loop Tranfer Function!! Bode magnitude criterion for calar open-loop tranfer function!! High gain at low input frequency!! Low gain at high input frequency!! Behavior of unit-gain cloed-loop tranfer function with high and low open-loop amplitude ratio = A( j ) 1+ A( j ) A( j )!y j!y C j # A( j )#0 # 1 A( j )# A( j! ) 22
Additive Variation in A() A o ( ) = C o ( I n! F o )!1 G o A( ) = A o ( ) +!A( ) Connection to LQ open-loop tranfer matrix Gain Change!A C ( ) =!C( I n F o ) 1 G o Control Effect Change!A G ( ) = C o ( I n F o ) 1!G!A F Stability Matrix Change 1 # & In ( F ) 1 { # o & } G o = C o I n ( F o +!F) 23 Conervative Bound for Additive Variation in A() Aume original ytem i table A o ( )! I m + A o ( ) # 1 Wort-cae additive variation doe not de-tabilize if! A( j# ) &' <! I m + A o j# &', 0 < # < ( Sandell, 1979 24
Bode Plot of Singular Value Singular value have magnitude but not phae 20 log!! # I m + A o ( j )&! A( j# )&' log! Stability guaranteed for changing! A j# &' up to the point that it touche! I m + A o j# &' 25 Multiplicative Variation in A() A o!1 G o ( ) = C o I n! F o A( ) = L PRE ( )A o ( ) or A( ) = A o ( )L POST ( ) Very complex relationhip to ytem equation; uppoe!l!l L( ) = I 3 +! # # # l 11 ( ) 0 0 0 0 0 0 0 0 & & & = I 3 + 'L( ) affect firt row of A o ( ) for pre-multiplication affect firt column of A o ( ) for pot-multiplication Sandell, 1979 26
! L( j# ) Bound on Multiplicative Variation in A() A o!1 G o ( ) = C o I n! F o &' <! I + A (1 m o j# & ', 0 < # < ) 20 log! 1! I m + A o j# & '! L( j# )&' log! 27 Deirable Bode Gain Criterion Attribute At low frequency! # I m + A( j )& >! min ( ) > 1! I m + A 1 1 j# & ' At high frequency { } = 1 <!! I m + A 1 max # j# & ' 28
Deirable Bode Gain Criterion Attribute! # A( j )& Undeirable Region Undeirable Region! # A( j )&! C1! C2 log! Croover Frequencie 29 Senitivity and Complementary Senitivity Matrice of A()! I m + A( ) S Senitivity matrix! # 1 Invere return difference matrix # I m + A!1 Complementary enitivity matrix T( )! A( ) I m + A( )! # 1 30
Senitivity and Complementary Senitivity Matrice of A() Small S( j! ) implie low enitivity to parameter variation a a function of frequency! I m + A( j! ) S j! # &1 Small T( j! ) implie low noie repone a a function of frequency T( j! )! A( j! ) I m + A( j! ) # &1 31! But Senitivity and Complementary Senitivity Matrice of A() S( j! ) + T( j! )! I m + A( j! ) = I m + A( j! ) # &1 + A j! # # I + A j! m &1 = I m I m + A( j! ) # &1! Therefore, there i a tradeoff between robutne and noie uppreion S( j! ) T( j! ) log! 32
Next Time:! Probability and Statitic! 33 Supplemental Material 34
Alternative Criteria for Multiplicative Variation in A()! Definition! OL!! OL! CL : Open-loop characteritic polynomial of original ytem : Perturbed characteritic polynomial of original ytem : Stable cloed-loop characteritic polynomial of original ytem {!! OL ( j ) = 0} implie that! OL ( j ) = 0 for any on # R (i.e., vertical component of D contour) = &' I m + A o j () for any on # R Lehtomaki, Sandell, Athan, 1981 35 Alternative Criteria for Multiplicative Variation in A()! L 1 ( j# ) I m & ' < ( =! I m + A o j# &' And at leat one of the following i atified: ( < 1 Perturbed cloed-loop ytem i table if + L( j# ) ) 0 L H j# 4 (( 2 1)! 2 L( j# ) I m &' > ( 2! 2 L( j# ) + L H ( j# ) 2I m & ' 36
If Guaranteed Gain and Phae Margin! # I m + A o j & > ' o ( 1! Guaranteed Gain Margin 1 K = 1 ±! o! Guaranteed Phae Margin! = ± co 1 # 2 ' o & 2 ( ) In each of m control loop 37 Guaranteed Gain and Phae Margin If L H ( j! ) + L( j! ) 0 and A H o ( j! ) + A o ( j! ) 0! Guaranteed Gain Margin! Guaranteed Phae Margin K = ( 0,! )! = ±90 In each of m control loop 38