Singular Value Analysis of LinearQuadratic Systems !
Robert Stengel!
Optimal Control and Estimation MAE 546 !
Princeton University, 2017
!! Multivariable Nyquist Stability Criterion
!! Matrix Norms and Singular Value Analysis
!! Frequency domain measures of robustness
!! Stability Margins of Multivariable Linear-Quadratic
Regulators
Copyright 2017 by Robert Stengel. All rights reserved. For educational use only.
http://www.princeton.edu/~stengel/MAE546.html
http://www.princeton.edu/~stengel/OptConEst.html
1
Scalar Transfer Function and Return
Difference Function
•! Unit feedback
control law
•! Block diagram
algebra
!y ( s ) = A ( s ) #$ !yC ( s ) " !y ( s ) %&
#$1+ A ( s ) %& !y ( s ) = A ( s ) !yC ( s )
!y ( s )
A(s)
"1
=
= A ( s ) #$1+ A ( s ) %&
!yC ( s ) 1+ A ( s )
A ( s ) : Transfer Function
!"1+ A ( s ) #$ : Return Difference Function
2
A(s) =
Relationship
Between SISO
Open- and ClosedLoop Characteristic
kn ( s )
Polynomials
! OL ( s )
!y ( s )
kn ( s ) ! OL ( s )
kn ( s )
=
=
!yC ( s ) "#1+ kn ( s ) ! OL ( s ) $% ! OL ( s ) "#1+ kn ( s ) ! OL ( s ) $%
kn ( s )
kn ( s )
=
=
! CL ( s )
"# ! OL ( s ) + kn ( s ) $%
!! Closed-loop polynomial is
open-loop polynomial
multiplied by return difference
function
! CL ( s ) = ! OL ( s ) "#1 + A ( s ) $%
3
Return Difference Function Matrix
for the Multivariable LQ Regulator
Open-loop system
s!x(s) = F!x(s) + G!u(s)
( sI " F ) !x(s) = G!u(s)
"1
!x(s) = ( sI " F ) G!u(s)
Linear-quadratic feedback control law
!u ( s ) = "R "1GT P!x ( s ) ! "C!x ( s )
= "C ( sI " F ) G!u(s) ! "A ( s ) !u(s)
"1
4
Multivariable LQ Regulator Portrayed
as a Unit-Feedback System
5
Broken-Loop Analysis of Unit-Feedback
Representation of LQ Regulator
Cut the loop as shown
Analyze signal flow from ! ( s ) to ! ( s )
!"" (s) = $% !uC (s) # A ( s ) !"" (s) &'
$% I m + A ( s ) &' !"" (s) = !uC (s)
!"" (s) = $% I m + A ( s ) &' !uC (s)
#1
!"u(s) = A ( s ) "## (s) = A ( s ) $% I m + A ( s ) &' "uC (s)
!1
Analogy to SISO closed-loop transfer function
6
Broken-Loop Analysis of Unit-Feedback
Representation of LQ Regulator
Cut the loop as shown
Analyze signal flow from ! u ( s ) to ! u ( s )
A ( s ) = C ( sI n ! F ) G
!1
!"u(s) = A ( s ) "## (s) = A ( s )[ "uC (s) + "u(s)]
! "# I m + A ( s ) $% &u(s) = A ( s ) &uC (s)
!&u(s) = "# I m + A ( s ) $% A ( s ) &uC (s)
!1
7
Closed-Loop Transfer Function
Matrix is Commutative
!"u(s) = A ( s ) #$ I m + A ( s ) %& "uC (s)
!1
!"u(s) = #$ I m + A ( s ) %& A ( s ) "uC (s)
!1
2nd-order example
A[I + A] = [I + A] A
!1
" a1
$
$# a3
a2 % " a1 + 1
a2
'$
a4 ' $ a3
a4 + 1
&#
" (a a ! a a + a )
1 4
2 3
1
=$
$
a3
#
!1
!1
%
" a1 + 1
a2
' =$
a4 + 1
'&
$# a3
1
( a1a4 ! a2 a3 + a4 )
%
'
'&
!1
" a1
$
$# a3
a2 %
'
a4 '
&
%
' det ( I 2 + A )
'
&
8
Relationship Between Multi-Input/MultiOutput (MIMO) Open- and Closed-Loop
Characteristic Polynomials
I m + A ( s ) = I m + C ( sI n ! F ) G
!1
= Im +
! OL ( s ) I m +
CAdj ( sI n ! F ) G
" OL ( s )
CAdj ( sI n " F ) G
= ! OL ( s ) I m + A ( s ) = ! CL ( s ) = 0
! OL ( s )
Closed-loop polynomial is open-loop polynomial multiplied by
determinant of return difference function matrix
9
Multivariable Nyquist
Stability Criterion!
10
Ratio of Closed- to Open-Loop
Characteristic Polynomials Tested in
Nyquist Stability Criterion
Scalar Control
" CAdj ( sI n & F ) G $
! CL ( s )
= "#1+ A ( s ) $% = '1+
(
! OL ( s )
! OL ( s )
#
%
= a ( s ) + jb ( s ) Scalar
Multivariate Control
CAdj ( sI n " F ) G
! CL ( s )
= Im + A ( s ) = Im +
! OL ( s )
! OL ( s )
= a ( s ) + jb ( s ) Scalar
11
Multivariable Nyquist
Stability Criterion
! CL ( s )
= I m + A ( s ) ! a ( s ) + jb ( s ) Scalar
! OL ( s )
Same stability criteria for encirclements of –1 point
apply for scalar and vector control
Eigenvalue Plot, “D Contour”
Nyquist Plot
Nyquist Plot, shifted origin
12
Limits of Multivariable
Nyquist Stability Criterion
! CL ( s )
= I m + A ( s ) ! a ( s ) + jb ( s ) Scalar
! OL ( s )
•! Multivariable Nyquist Stability Criterion
–! Indicates stability of the nominal system
–! In the |I + A(s)| plane, Nyquist plot depicts
the ratio of closed-to-open-loop
characteristic polynomials
•! However, determinant is not a good
indicator for the size of a matrix
–! Little can be said about robustness
–! Therefore, analogies to gain and phase margins
are not readily identified
13
Determinant is Not a Reliable
Measure of Matrix Size
! 1 0 $
A1 = #
&;
0
2
"
%
! 1 100 $
A2 = #
&;
0
2
"
%
A1 = 2
A2 = 2
! 1 100 $
A3 = #
&;
0.02
2
"
%
A3 = 0
•! Qualitatively,
–! A1 and A2 have the same determinant
–! A2 and A3 are about the same size
14
Matrix Norms and
Singular Value Analysis!
15
Vector Norms
Euclidean norm
( )
x = xT x
1/2
Weighted Euclidean norm
(
Dx = xT DT Dx
)
1/2
For fixed value of ||x||,
||Dx|| provides a measure of the size of D
16
Spectral Norm (or Matrix Norm)
Spectral norm has more than one size
D = max Dx is real-valued
x =1
dim ( x ) = dim ( Dx ) = n ! 1; dim ( D ) = n ! n
Also called Induced Euclidean norm
If D and x are complex
x = (xH x)
1/2
Dx = ( x H D H Dx )
1/2
x H ! complex conjugate transpose of x
= Hermitian transpose of x
17
Spectral Norm (or Matrix Norm)
Spectral norm of D
D = max Dx
x =1
DTD or DHD has n eigenvalues
Eigenvalues are all real, as DTD is symmetric and DHD is
Hermitian
Square roots of eigenvalues are called singular values
18
Singular Values of D
Singular values of D
! i ( D ) = "i ( DT D ) , i = 1, n
Maximum singular value of D
! max ( D ) ! ! ( D ) ! D = max Dx
x =1
Minimum singular value of D
! min ( D ) ! ! ( D ) = 1 D "1 = min Dx
x =1
19
Comparison of
Determinants and
Singular Values
! 1 0 $
A1 = #
&;
" 0 2 %
! 1 100 $
A2 = #
&;
" 0 2 %
A1 = 2
A2 = 2
! 1 100 $
A3 = #
&;
" 0.02 2 %
A3 = 0
•!
•!
•!
Singular values provide a better
portrayal of matrix size, but ...
Size
is multi-dimensional
Singular values describe
magnitude along axes of a multidimensional ellipsoid e.g.,
A1 : ! ( A1 ) = 2; ! ( A1 ) = 1
x 2 y2
+
=1
!2 !2
A 2 : ! ( A 2 ) = 100.025; ! ( A 2 ) = 0.02
A 3 : ! ( A 3 ) = 100.025; ! ( A 3 ) = 0
20
Stability Margins of
Multivariable LQ Regulators!
21
Bode Gain Criterion and the
Closed-Loop Transfer Function
!! Bode magnitude criterion for
scalar open-loop transfer function
!! High gain at low input frequency
!! Low gain at high input frequency
!! Behavior of unit-gain closed-loop
transfer function with high and low
open-loop amplitude ratio
A ( j" )
!y ( j" )
=
!yC ( j" ) 1+ A ( j" )
A ( j! )
$$$$
# A ( j" )
A( j" ) #0
$$$$
#1
A( j" ) #%
22
Additive Variations
in A(s)
A o ( s ) = C o ( sI n ! Fo ) G o
!1
A ( s ) = A o ( s ) + !A ( s )
Connections to LQ open-loop transfer matrix
Gain Change
!A C ( s ) = !C ( sI n " Fo ) G o
Control Effect Change
!A G ( s ) = C o ( sI n " Fo ) !G
{
"1
"1
Stability Matrix Change
!A F ( s ) = C o #$ sI n " ( Fo + !F ) %& " #$ sI n " ( Fo ) %&
"1
"1
}G
o
23
Conservative
Bounds for Additive
Variations in A(s)
Assume original system is stable
A o ( s ) !" I m + A o ( s ) #$
%1
Worst-case additive variation does not de-stabilize if
! $% "A ( j# ) &' < ! $% I m + A o ( j# ) &' , 0 < # < (
Sandell, 1979
24
Bode Plot” of Singular Values
Singular values have magnitude but not phase
! #$ I m + A o ( j" ) %&
20 log !
! $% "A ( j# ) &'
log !
Stability guaranteed for changing ! $% "A ( j# ) &'
up to the point that it touches ! $% I m + A o ( j# ) &'
25
Multiplicative
Variations in A(s)
A o ( s ) = C o ( sI n ! Fo ) G o
!1
A ( s ) = L PRE ( s ) A o ( s ) or A ( s ) = A o ( s ) L POST ( s )
Very complex relationship to system equations; suppose
! l (s) 0 0 $
# 11
&
L ( s ) = I3 + # 0
0 0 & = I 3 + 'L ( s )
# 0
0 0 &%
"
!L ( s ) affects first row of A o ( s ) for pre-multiplication
!L ( s ) affects first column of A o ( s ) for post-multiplication
Sandell, 1979
26
Bounds on Multiplicative
Variations in A(s)
A o ( s ) = C o ( sI n ! Fo ) G o
!1
! $% "L ( j# ) &' < ! $% I m + A (1
o ( j# ) &
', 0 < # < )
! $% I m + A "1
o ( j# ) &
'
20 log !
! $% "L ( j# ) &'
log !
27
Desirable Bode Gain Criterion
Attributes
At low frequency
! #$ I m + A ( j" ) %& > ! min (" ) > 1
{
At high frequency
! $% I m + A "1 ( j# ) &'
"1
}
=
1
< ! max (# )
! $% I m + A "1 ( j# ) &'
28
Desirable Bode Gain
Criterion
Attributes
! #$ A ( j" ) %&
Undesirable
Region
Undesirable
Region
! #$ A ( j" ) %&
! C1
! C2
log !
Crossover Frequencies
29
Sensitivity and Complementary
Sensitivity Matrices of A(s)
Sensitivity matrix
S ( s ) ! !" I m + A ( s ) #$
%1
Inverse return difference matrix
"# I m + A !1 ( s ) $%
Complementary sensitivity matrix
T ( s ) ! A ( s ) !" I m + A ( s ) #$
%1
30
Sensitivity and Complementary
Sensitivity Matrices of A(s)
Small S ( j! ) implies low sensitivity to parameter variations as a function of frequency
S ( j! ) ! "# I m + A ( j! ) $%
&1
Small T ( j! ) implies low noise response as a function of frequency
T ( j! ) ! A ( j! ) "# I m + A ( j! ) $%
&1
31
Sensitivity and Complementary
Sensitivity Matrices of A(s)
•! But
S ( j! ) + T ( j! ) ! "# I m + A ( j! ) $% + A ( j! ) "# I m + A ( j! ) $%
&1
&1
= "# I m + A ( j! ) $% "# I m + A ( j! ) $% = I m
&1
•! Therefore, there is a
tradeoff between
robustness and
noise suppression
S ( j! )
T ( j! )
log !
32
Next Time:!
Probability and Statistics!
33
Supplemental Material
34
Alternative Criteria for
Multiplicative Variations in A(s)
•! Definitions
! OL ( s ) : Open-loop characteristic polynomial of original system
!! OL ( s ) : Perturbed characteristic polynomial of original system
! CL ( s ) : Stable closed-loop characteristic polynomial of original system
{!!
OL
( j" ) = 0} implies 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, Athans, 1981
35
Alternative Criteria
for Multiplicative
Variations in A(s)
Perturbed closed-loop system is stable if
! $% L"1 ( j# ) " I m &' < ( = ! $% I m + A o ( j# ) &'
And at least one of the following is satisfied:
• ( <1
• LH ( j# ) + L ( j# ) ) 0
(
)
• 4 ( 2 " 1 ! 2 $% L ( j# ) " I m &' > ( 2! 2 $% L ( j# ) + LH ( j# ) " 2I m &'
36
Guaranteed Gain and
Phase Margins
If
! #$ I m + A o ( j" ) %& > ' o ( 1
•! Guaranteed Gain Margin
K=
1
1 ± !o
•! Guaranteed Phase Margin
In each of m
control loops
$ # o2 '
! = ± cos & 1 " )
2 (
%
37
Guaranteed Gain
and Phase Margins
If LH ( j! ) + L ( j! ) " 0 and A o H ( j! ) + A o ( j! ) " 0
•! Guaranteed Gain Margin •! Guaranteed Phase Margin
K = ( 0, ! )
! = ±90°
In each of m control loops
38