5/12/2016
Batch Calculation of the
Residues and Their Sensitivities
Or: How to compute almost any derivative using
>>sum(prod([combnk(factors), dfactors.’]))
Bradley T. Burchett
Outline
• Motivation - Brute force Control System
Optimization
• Sylvester’s Expansion
• Partial Fraction Expansion - Batch Method
• Numerical Examples
• Derivatives of the Residues
• Numerical Examples
• Conclusions
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Motivation: Brute Force
Optimization of Control Systems
• Infinite horizon optimal linear feedback controllers are
determined by minimizing the cost function
• Subject to the constraints defined by the system state
dynamics
Motivation continued
• In the case of State Feedback, there is a direct solution from
the Algebraic Ricatti Equation
• In the case of Output feedback, only iterative solutions are
available, the most historic involving a pair of Lyapunov
equations.
• Linear dynamics always have a closed-form solution.
• Idea: substitute the closed-loop system closed-form solution
into the cost function to transform the optimization into an
unconstrained optimization.
• The reformulated output feedback problem is termed ‘brute
force optimization of control systems’
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A Plethora of methods
• The closed-form solution of the state dynamics for any
time is written in terms of the matrix exponential of At
• Thus for every method of computing the matrix
exponential, there exists a corresponding brute force
optimization method
• Investigations to date have used the following methods
– Dyadic decomposition (Burchett, Costello 1997)
– Pade Approximation (Burchett, Costello 2001)
– Sylvester’s expansion (current unpublished work)
Sylvester’s expansion
• Sylvester’s expansion for systems with roots of
multiplicity mk is written as:
• nk is the combined numerator polynomial from the partial
fraction expansion of:
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• Using Sylvester Expansion, the quadratic cost function can
be re-written
• Where
•
Φ(Α
Α) is the minimum polynomial of A
• The cost function in this form can be integrated closedform yielding (this requires integration by parts and
invoking mathematical induction)
• We would like to invoke gradient based methods with
analytic derivatives if possible
• This requires computing the derivatives of all factors in the
equation above (including eigenvalues and residues).
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Partial Fraction Expansion
• The partial fraction expansion of a system transfer function
• Can be written
• Adding the RHS of the previous eqn by finding a common
denominator then equating powers of s in the numerators
right and LHS of the previous eqn, a set of n linear
equations emerges. This set of equations can be written in
a matrix form
Ξ a=H
• H is a vector of coefficients from the convolved form of
the original numerator polynomial
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•
•
Ξ has a distinct pattern in terms of row i and column j.
In Matlab code, that is
»Xi(i,j)=sum(prod(combnk(-poles([1:j-1, j+1:n]),i-1)));
• For the distinct poles case:

1


m −1

∑ − pj

j=2


 ∑ − p j (− pk )


 jk =  m2−1
jk
≠1


M


m −1

− pj
∏

j=2

( )
L
1
m −1
∑−p
L
j
j =1
j≠2
∑ (− p )(− p )
j
 m−1 
jk = 

 2 
jk ≠2
k
M
L
M
m −1
∏−p
j =1
j≠2
j
L



m −2

∑ −pj

j =1

  n1   η1 
− ∑ − p j (− pk )  n2   η 2 
 m −1 
=
jk =

 M   M 
 2 


jk ≠m −1
 n  η 
m −1
m −1

M


m−2

−
p
∏
j
j =1


1
( )
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• For systems with repeated poles, the PFE is:
• By repeating the algebraic process outlined in above, we
discover that the resulting system of equations Ξ a=H has
the same form as Eq. 4, except that for the terms involving
repeated poles, the corresponding column of the Ξ matrix
can be built from the bottom up using Eq. 4, pretending
that the system is lacking mk-j+1 occurrences of the
repeated pole.
• Or more specifically:
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Numerical Examples
• First case: distinct poles:
• The PFE is:
• The matrix equation is
• Where
• The residues are a1,2=0.107±0.063j, a3,4=0.274±0.296j
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• Second case: Repeated poles
• The PFE is:
• The matrix equation is
• The resdiues are a1,2=-0.25, a3,4±0.25j
Derivatives of the Residues
• Since we have constructed the residue calculation as a
linear system
a=Ξ
Ξ-1H
• The derivatives are found by application of the product
rule:
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• Then using the matrix inversion fact
• We obtain
• Which is good news indeed--we do not require the inverse
of the matrix derivative
• For the distinct poles case, the matrix derivative
follows from the original pattern
n
 ∂p 
= ∑  − m ∏(− pk )
m =1 
∂K
∂ K  k = n −2 
∂ Ξi , j
m≠ j
 i −2 


k≠m
• In Matlab code, if
»rows = combnk(-poles([1:j-1, j+1:n]),k)
;
»column = fliplr(-dpoles(1:j-1, j+1:n));
• Then
»dXi = sum(prod(combnk(rows(m,:),k-1), column(:),2));
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• More explicitly:
0

n−1
∂pj

∑− ∂K

j=2

 ∂p 
 ∑  − j  (− pk )
n −1
 j ,k = P2  ∂ K 
j, k ≠1

M
 n  ∂p m 
−
∏ (− pk )
∑
m=1 
∂K  k = n− 2 
 i− 2 
m≠1


k≠m

L
0
n−1
∂pj
∑− ∂K
L
j =1
j≠2
 ∂pj 
−
 (− pk )
n−1
∂K 
j ,k = P2 
∑
L
j ,k ≠ 2
M
 ∂p m 
−
∑
m =1 
∂K 
n
m≠2
M
∏(− p )
 n− 2 

k =
 i −2 
k≠m


∑− ∂K

j =2

 ∂p j 


−
−
p

(
)
∑n−1
k
∂K 
j, k = P2 

j ,k ≠ n

M
n

 ∂p m 
−
− pk )
(
∑
∏


m=1
∂K k = n− 2 
 i −2 
m ≠n



k≠ m

0
n
k
L
∂pj
Numerical Example
• Considering the Frequency domain function
• Assuming the pole derivatives are known as
• The derivative matrix is
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• Where
• The derivative of the H matrix is taken to be
∂H
∂K
= [0
0
0
0 1]
• The resulting residue derivatives are:
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Repeated Poles Example
• Consider again the system
• With pole derivatives
• This leads to an enigma
– when constructing the columns of the Ξ matrix
corresponding to the repeated pole, we ignored
one occurrence of the repeated pole
– When constructing the derivative of the Ξ matirx,
which pole derivative to I ignore?
• I can show what the numbers should be, from
finite differencing, but I cannot determine the
correct pattern...
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• The derivative of the Ξ matrix is
• The residue sensitivities are
Conclusions
• Batch calculation seems to be a viable
method for finding derivatives of the
residues
• The important case of systems with
repeated poles needs further investigation
• Help ?!
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