Determining the Seismic Transfer Function Infimum for a Structural
Design
C. E. Hann & J. G. Chase
Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
W.-H. Wu
Department of Construction Engineering, National Yunlin University of Science and Technology, Taiwan
ABSTRACT: The overall seismic structural design problem, for a given structure, may be considered in terms
of its transfer function from the ground motion input to the response output. In this regard, it is the transfer
function infimum, or the greatest lower bound, over all possible ground motions that is of interest, as it defines the maximum structural response to any ground motion. More importantly, this infimum value can
change over time as damage occurs, and tracking it in real-time would provide significant health monitoring
information. However, determining this value, especially for large or complex models, is computationally intense and numerically very ill-conditioned. This research presents a highly efficient, stable and computationally rapid method for determining the seismic transfer function infimum for any structural model undergoing a
ground motion. This method is based on the Routh-Hurwitz criterion and provides a simple semi-analytical
approach, enabling real-time computation for a given model.
1 INTRODUCTION
The H  control formulation was first introduced
in (Zames 1981), and limits the infinity norm of the
transfer function between disturbance inputs and
regulated outputs to a value  . This formulation has
been applied in many fields, including active structural control (Chase et al. 1996,Yang et al. 2004).
An important consideration in the design of H 
controllers is the optimal H  norm, or the infimum
of the H  optimal control problem (denoted   in
this paper). The computation of this infimum has
typically been studied based on either iterative
(Doyle et al. 1989, Gahinet 1994, Lin et al. 2000,
Stoorvogel 1992, Gahinet and Apkarian 1994) or
non-iterative methods (Chen 1997, Chu 2004).
However, the algorithms are computationally expensive and can be numerically ill-conditioned when  is
close to   ( Chen 1997).
In this paper, a computationally efficient iterative
algorithm is developed for the determination of  
by taking a completely novel approach. First, the eigenvalues of the Hamiltonian matrix associated with
the ARE problem are examined to define a borderline stability criterion associated with the H  problem infimum. Based on this stability criterion, the
classical Routh-Hurwitz theorem is employed to
check the system stability, requiring only the characteristic polynomial coefficients of the Hamiltonianmatrix for any given value of  . Moreover, it is
shown that the characteristic polynomial can be analytically expressed in terms of  and thus used to
economically obtain the polynomial coefficients required in the iteration process corresponding to various values of  .
Thus, a novel Routh-Hurwitz based method to
compute the optimal H  norm can then be established. An 8-DOF numerical example is used to validate the effectiveness and efficiency of the new
method. The closed-form solution to single-degreeof-freedom (SDOF) structural control problem (Wu
& Lin 2004) is employed to further validate the
method.
2 PROBLEM AND STABILITY CRITERION
2.1 Problem statement
Consider the standard linear time-invariant (LTI)
system defined:
x  Ax  Bu  Ew
z  Cx  Du
(1)
where the state x  R n , the control input u  R m , the
disturbance w  R l , and the regulated output
z  R p . In addition, A, B, E, C and D are constant
matrices of appropriate dimension. If state feedback,
u  Gx is considered, the closed-loop system is defined:
x  (A  BG)x  Ew
(2)
z  (C  DG) x
The H  norm of this system, S, is defined in the time
domain as
S

 sup
w
z
w
2
(3)
2
Therefore, the infimum of the H  norm for S under
state feedback can be defined:

   inf S

G  R mn , A cl is stable

(4)
where A cl is the closed-loop plant matrix. In other
words, the optimal norm   is the minimum  value
(or threshold) for which controlled system stability
can be guaranteed, given the problem described in
Equation (1).
For a given suboptimal     , the corresponding
H  control problem is to determine the state feedback control gain matrix G such that
S

 sup
w
z
w
2


  or 0 z T z dt   2 0 w T w dt
(5)
2
To mathematically analyze the H  control problem,
a quadratic performance index J is usually defined
and Equation (5) can be further reformulated:
1
min max J  min max
u
u
w
w 2


z
T

z  γ 2 w T w d t  0 (6)
0
Equation (6) is a minmax problem under the state
motion constraint of Equation (1). Calculus of variation has been applied to solve this constrained optimization problem and the resulting solution takes the
form of an algebraic Ricccati equation (ARE) (Doyle
et al., 1989). In most practical applications, it is
normally further assumed that C T D  0 and D T D is
full rank. With these conditions, the original H 
ARE can be defined:

 1
PA  A T P  P 2 EE T  B D T D
 

1

BT P  C T C  0

(7)
where P  P T  0 is the positive-definite Riccati matrix. The resulting closed-loop system matrix is thus
defined:

 1
A cl  A   2 EE T  B D T D
 

1

BT P

(8)
To obtain the n  n matrix P , it is most convenient
to transform Equation (7) into a linear eigenvalue
problem for the Hamiltonian matrix, H.

1

A
EE T  B D T D

2
H


T
 AT
 C C

1

BT 


(9)
P can then be directly obtained from the eigenvalues
and eigenvectors of H (Meirovitch, 1990).
2.2 Stability criterion
Since Equation (7) is a quadratic matrix equation,
more than one solution for P can be obtained. To
guarantee the stability of the controlled system, a
positive semi-definite or better, symmetric solution,
P  P T  0 , is required.
A numerically efficient and novel, stability criterion is adopted in this study by examining the eigenvalues of the Hamiltonian matrix in Equation (9).
This matrix is known as it is only composed of a
given  value and constant matrices A, B, C, D and
E that define the problem. Since H is a 2n  2n matrix, there are 2n eigenvalues with 2n corresponding
eigenvectors. It is clear that all the eigenvalues of the
closed-loop system matrix A cl are included in those
of H with the transformation from Equation (7) to
(9) (Meirovitch, 1990). Moreover, Potter (1966) also
proved that the eigenvalues of the Hamiltonian matrix
H
appear
in
anti-symmetric
pairs
 1 ,  2 , ,  n in the complex plane.
Following from these attributes, there are only n
eigenvalues with negative real parts for H and these
stable eigenvalues have to be selected in the determination of P to find a stable closed loop solution
for a given value of . In addition, these eigenvalues
move in the complex plane as the value of  is
changed, while retaining anti-symmetry. Therefore,
for a stable closed loop solution, no value of  can be
chosen that results in one or more pure imaginary,
borderline stable eigenvalues of H.
Thus, a simple stability criterion for the H  infimum can be established by prohibiting values of 
for which eigenvalues of H are located on the imaginary axis. More specifically, as     the eigenvalues of H move symmetrically towards the imaginary axis. The infimum,   , is the value of  where
the first eigenvalues become purely imaginary valued.
With this stability criterion, the classical RouthHurwitz theorem (Hurwitz, 1964) can be readily
employed to check the stability of a H  controlled
system for any given value of . More specifically,
creating a Routh table the first column can be
checked for zero values indicating purely imaginary
borderline stable eigenvalues in the characteristic
polynomial for H. The infimum,   is thus the
smallest value of for which a zero appears in the
first column, or equally simply for which the product
of the first column becomes zero.
3 ROUTH-HURWITZ APPROACH
In this section, an analytical method is developed
for the evaluation of the characteristic polynomial
that utilizes the structure of the Hamiltonian matrix,
H, and requires minimal computation. The RouthHurwitz theorem provides an expedient procedure to
test the stability of a system merely from the coefficients of its characteristic polynomial, without evaluating any eigenvalues. Therefore, the computation
to solve ARE’s or LMI’s in other iterative methods
can be avoided. Instead, simple calculations using
only the polynomial coefficients are required with
the application of Routh-Hurwitz theorem. Finally,
using the Routh-Hurwitz theorem, transforms the
problem into a simple interpolation problem requiring a limited number of Routh table evaluations.
3.1 Evaluating characteristic polynomial of H
The Hamiltonian matrix H given by Equation (9)
can be rewritten in the form:
 A  A1  A 2 
H
 A T 
A 3
(10)
where

A1  EE T , A 2   B D T D

1
B T , A 3  C T C,  
1
.
2
(11)
If the matrices A1 and A 3 are of rank r1 and r3 , respectively, then the coefficients of the characteristic
polynomial of H can be shown to be polynomials in
 of order r  min r1 , r3  at most.
Theorem 1: Given the 2n  2n matrix H defined by
Equation (10) where the minimum rank of the n  n
matrices A1 and A 3 is r, the characteristic polynomial of H can be written:
2n
char( H)  det( H  I)   pi ( )i
system A, the condition of det A  I   0 can be
obtained such that Equation (13) is further reduced
to
det( R  A)  0
(14)
where
R   A 3 A  I 1 A1 ,


(15)
A   A T  I  A 3 A  I 1 A 2
From
Equation
(15),
it
is
obvious that
rank( R )  min[rank( A1 ), rank( A3 )]  r
and from
Equation (13), it suffices to show that det  R  A  is
an order r polynomial in  . This is easily shown using the standard expansion of the determinant, thus
completing the proof. █
Based on Theorem 1, the coefficients pi ( ) of the
characteristic polynomial of H have to be polynomial functions of  and can consequently be predetermined. More specifically, the coefficients of
each polynomial pi ( ) in Equation (12) can be
uniquely solved from r  1 numerical calculations of
the characteristic polynomial corresponding to r  1
different selected values of  . Once computed,
these coefficients can be used to rapidly evaluate the
characteristic polynomial for any  .
3.2 Routh Hurwitz stability criterion
Considering the anti-symmetric eigenvalues of H,
the polynomial of Equation (12) has to be even.
Hence, a special case of the Routh-Hurwitz stability
criterion and table must be applied (Gantmacher
1959) to fill out the second row. Following
(Gantmacher 1959) and given a characteristic polynomial of H:
q( )  2n  a 2n 2 ( )2n 2    a 2 ( )2  a 0 ( )
(16)
(12)
i 0
and the derivative q ' ( ) is defined:
Proof (outline): The characteristic polynomial of H
can be reformulated to create a new condition.
q' ( )  2n2n 1  (2n  2)a 2n ( )2n 3    2a 2 ( )
(17)
 A  I  A1  A 2 

char(H )  det 
 A T  I 
 A3
 det A  I 


The first two Routh table rows are then defined:

 det  A  I  A 3 A  I   A1  A 2 
T
1
0
(13)
Since the eigenvalues of the closed-loop system matrix A cl are not the same as those of the open-loop
r1  1 a 2n 2
r2  2n
 a2
( 2n  2) a 2 n  2
a0 
 2a 2
0
(18)
It should be noted that the coefficient of 2 n is chosen as 1 without losing any generality. Defining:
b1,1  1, b1,2  a 2n 2 , , b1,n  a 2 , b1,n 1  a 0
b2,1  2n, b2,2  (2n  2)a 2n 2 , , b2,n  2a 2 , b2,n 1  0
(19)
From Equation (19), rows r3 ,  , r4 n of the Routh
Table are defined:


r3  b3,1 b3,2  b3,n b3, j  


1 b1,1
b2,1 b2,1
r4  b4,1 b4,2  b4,n 1 b4, j  

r5  b5,1
b5,2
b2, j 1
1 b2,1
b3,1 b3,1

, j  1,  , n
b2, j 1
b3, j 1
1 b3,1
b4,1 b4,1
b5,n 1 b5, j  

b1, j 1
b3, j 1
b4, j 1
,
j  1,  , n  1

b4 n  2 , j  

b4 n  4, j 1
b4 n  3,1 b4 n  3,1
b4 n  3, j 1
r4 n 1  b4 n 1,1 b4 n 1,2
b4 n 1, j  


b4 n  4,1
1

b4 n  3,1
b4 n  3, j 1
b4 n  2,1 b4 n  2,1
b4 n  2, j 1
1

r4 n  b4 n ,1 , b4 n ,1  
,
j  1, 2
j  1, 2
b4 n  2,1
b4 n  2 , 2
b4 n 1,1 b4 n 1,1
b4 n 1,2
1
.
(20)
The value of  * may then be computed by looking
for zero values in elements of the first column,
where the pre-defined first two entries need not be
considered. As discussed earlier, the first zero of individual elements in the first column will also show
up as a first zero in the product of the elements, assuming the non-degenerate case of only one element
crossing a zero value at any one time. Therefore, this
product can be defined:
  b3,1    b4n,1
(21)
To find the first zero in  , let  L and  U correspond
to values of  where  ( L ) ( U )  0 and  L   U .
Then  L   *   R , where the infimum is given as:
 
*  *
1 / 2
Step 1: Obtain A,B,C,D,E from Equation (1).
Step 2: Precompute all coefficients of pi ( ) in
Equation (12) by solving the equations generated by
r  1 numerical calculations of the characteristic polynomial corresponding to r  1 different selected
values of  .
Step 3: Choose  L and  U that correspond to
 ( L ) ( U )  0 and  L   U , where  is given by
Equation (21).
Step 4: Choose N values of    i   L ,  R  . For
each  i , i  1, , N , compute the Routh table for the
characteristic polynomial of the Hamiltonian matrix
H given by Equation (16) and then compute  i using
Equation (21).

r4 n  2  b4 n  2,1 b4 n  2,2
3.3 Algorithm
(22)
The value in Equation (22) can be approximated by
choosing N values of    i   L ,  R  with corresponding values of    i , i  1, , N calculated
from Equation (21). Fitting these points
 1 , 1 , ,  N ,  N  with a least squares polynomial f of order less than N creates an approximation of
 * that can be used to effectively eliminate further
iterations, an analytical approximation of the solution. Thus, the method results in an accurate approximation to  * using only those N evaluations required to obtain the polynomial to interpolate the
final solution. In addition, only limited computational intensity is required to obtain each point.
Step 5: Fit a least squares polynomial f of order
less
than
N
through
the
N
points
 1 , 1 , ,  N ,  N  calculated in Step 4 and compute the zero crossing  * , an approximation to the
optimal H  norm using Equation (22).
Step 6: Output an approximation to the optimal H 
norm from Equation (22).
4 NUMERICAL VALIDATION
This section presents numerical examples from the
area of structural control to demonstrate the methods
presented. The initial case uses a previously published analytical solution to validate the accuracy of
the method.
4.1 Example 1: SDOF structural control case
A typical SDOF structural system is first taken as
a demonstrative example to illustrate the RouthHurwitz based method developed in this study. With
mass m, stiffness k, and damping c, its natural frequency and damping ratio are defined:   ( k / m)
and   c /(2m) . If C and D are defined corresponding to the H  energy control case in the literature
(Wu and Lin 2004):
0
C  0

0

0 


m  and D  

0 



0 
0 
1 

k
(23)
where  is a user specified energy weighting parameter. The associated Hamiltonian matrix H can
then be expressed:
 0

2
 
H
 0

 0
0




  1

 2 0

1

m   2

2

0
0


 m  1
2

1
0
2
(24)
If the earthquake problem with a single excitation
input is considered and one control input is exerted
on the top floor, as from an active mass damper, the
matrices B and E in Equation (27) are defined:
 1 
B
 and
0 71 
Going through the process analytically, the product
of the elements in the first column of the Routh Table is defined:
E  181
(30)

 1

δ  a 2 2  4 4   4 2  4 2    2  1  4 4 (25)





where 1 is a matrix of all ones.
Steps 1 to 5 of the algorithm given in Section 3.3
are
now
applied.
In
this
example,
rankA1   rank EET  1 , rank A 3   rank  CT C  8 .
Based on Theorem 1, it is clear that the coefficients
of the characteristic polynomial of H given by Equation (12) will be linear in  , defined:
The condition that  equals zero thus leads to:
char(H )  16  p14 ( )14    p 2 ( ) 2  p 0 ( )
2
 

  4
(26)
2
Equation (26) exactly matches that derived by separate means in Wu and Lin (2004), indicating that this
method matches the exact analytical result.
4.2 Example 2: 8-DOF structural control case
A second structural control case consisting of an
eight-story shear building was used by Wu and Tsai
(2006). It is considered here to investigate more
complex systems where an analytical solution does
not exist. Each floor has a mass of 345.6 tons and a
horizontal column stiffness of 340,400 kN/m, resulting in a first-mode frequency of 0.921 Hz. The
damping coefficient of each floor is 2,937 tons/sec,
corresponding to a first-mode damping ratio of
2.5%.
Following Wu and Tsai (2006), the matrices of
Equation (9) are defined:
I 88  T
0 88
 0
A   881
1 , C C  
 M K  M C 
0 88
 0

 0

D T D  1, B   811 , E   811 
 M B
 M E 
0 88 
M 
(27)
In Equation (27), the corresponding mass, stiffness
and damping matrices are:
M  345.6I 88 , K  340400 T88 , C  2937 T88 (28)
where
T88
0
0
0
0
0
 1 1 0
 1 2  1 0
0
0
0
0


 0 1 2 1 0
0
0
0


0
0 1 2 1 0
0
0

0
0
0 1 2 1 0
0


0
0
0 1 2 1 0 
0
0
0
0
0
0  1 2  1


0
0
0
0
0  1 2 
 0
(29)


 16  ( c14  d 14 )14  
(31)
 ( c 2  d 2  ) 2  ( c 0  d 0  )
The coefficients of c i and d i are calculated by computing two sets of the characteristic polynomial coefficients in Matlab for two values of   0 and   1 .
Given numerical values of p14 (0), p12 (0), , p0 (0)
and p14 (1), p12 (1), , p0 (1) from Equation (12), the
coefficients in Equation (31) can be obtained:
c 2i  p 2i (0),
d 2i  p 2i (1)  p 2i (0), i  0,  , 7
(32)
Figure 1 shows the product of the first column of
the Routh table,  , plotted against an extensive
range of  . To avoid numerical overflow in the calculation of  ( ) , for each  , all the elements in the
first column of the associated Routh table is divided
by the corresponding elements in the first column of
the Routh table for   0 . This effectively replaces  ( ) by  ( ) /  (0) .
Although there are several zero crossings in Figure 1, only the first (lowest value) is of interest. In
this case, values of  L and  R in step 3 of the algorithm are chosen to be  L  0 and  R  20 . In step 4,
N  5 values of  i are chosen to be  1  0,  2  5,
 3  10,  4  15 and  5  20 . The corresponding values of 1 to  5 are shown in Figure 2 along with the
interpolating polynomial, which is cubic.
Note that a linear interpolation for this case would
likely have been more than sufficient for this narrow
range of . In practice, a linear followed by a quadratic and so on up to higher order polynomials could
be tested by computing the least squares error.
Once the degree of polynomial is found that has a
least squares error less than a chosen tolerance, this
polynomial could be used as the interpolating polynomial. The first real root of the cubic corresponds
to the required zero crossing and is   8.4841 .
To check that this value is correct, the eigenvalues
of H are computed for   8.4841 and   8.4842 .
This test results in eigenvalues of  0.0003  5.7915 i
and  5.7911 i,  5.7919 i respectively. Thus, the infimum,
is
in
the
range
of
 ,
0
(8.4841) 2  0.013892     (8.4842) 2  0.013893 .
Therefore, the approximation of    0.01389 is accurate to within an absolute relative percentage error
of 0.01 % .
100
50
0

solutions required. As a result, the approach provides the desired result with minimum computation
compared to other approaches in the literature.
Two test cases are presented, including an 8-DOF
structural control case with an error within 0.01% of
iterative eigenvalue solutions.
Overall, the methods and theory presented comprise a Routh-Hurwitz based semi-analytical minimal iteration approach for determining the H  norm
infimum of a control system, and are a significant
step forward in this area of work.
-50
-100
6 REFERENCES
-150
-200
-250
0
0.5

1
1.5
2
4
x 10
Figure 1. Product of the first column of the Routh
table versus  for example 2. The optimal value is
very near 0 on the x-axis of this scale
1
0.5

0
-0.5
-1
-1.5
0
5
10
15
20

Figure 2. Plot of the five points  1 ,1 , ,  5 , 5 
for example 2 and the least squares interpolating
polynomial. The x-axis is much reduced from Figure
2.
5 CONCLUSIONS
A new method for determining the optimal H 
norm, or infimum, of a closed loop system has been
developed and presented. The new method is computationally far less intense as it does not require repeated solution of eigenvalues or matrix Riccati
equations. The method is based on the application of
Routh-Hurwitz theorem and the application of the
classical Routh table to check a stability condition
on the Hamiltonian matrix that is associated with the
infimum value. In addition, a method of interpolating to obtain an approximation of the optimal result
is presented that reduces the number of Routh table
Chase, J. G., Smith, H. A. & Suzuki, T. 1996. Robust H  control considering actuator saturation II: applications. J. Eng.
Mech., 122:984-993.
Chen, B. M. 1997. Exact computation of infimum for a class of
continuous-time H  optimal control problem with a nonzero direct feed through term from the disturbance input to
the controlled output. System and Control Letters, 32:99109.
Chu, D. 2004. On the computation of the infimum in H  optimization. Numer. Linear Algebra Appl., 11:619-648.
Doyle, J. C., Glover, K., Khargonekar P. P., & Francis, B. A.
1989. State-space solutions to standard H 2 and H  control
problems. IEEE Transactions on Automatic Control,
34:831-847.
Gahinet, P. 1994. On the game Riccati equations arising in
H  control problems. SIAM J. Control and Optim.,
32:635-647.
Gahinet, P. and Apkarian, P. 1994. A linear matrix inequality
approach to H  control. Int. J. Robust Nonlinear Control,
4:421-448.
Gantmacher, F. 1959. The Theory of Matrices, Vol II, New
York: Chelsea.
Hurwitz, A. 1964. On the conditions under which an equation
has only roots with negative real parts. Rpt. in Selected Papers on Mathematical Trends in Control Theory, R. T.
Ballman et al., Ed., New York: Dover.
Lin, W.-W. Wang C.-S. & Xu, Q.-F. 2000. Numerical computation of the minimal norm of the discrete-time output feedback control problem. SIAM Journal on Numerical Analysis, 38:515-547.
Meirovitch, L. 1990. Dynamics and Control of Structures. New
York: Wiley.
Stoorvogel, A. A. 1992. The Control Problem: A State Space
Approach. Englewood Cliffs: Prentice-Hall, 1992.
Wu W.-H. & Lin, C.-C. 2004. H  energy control and its stability analysis for civil engineering structures. Struct. Control Health Monitoring, 11:161-187.
Yang, J. N., Lin S. & Jabbari, F. 2004. H  -based strategies
for civil engineering structures. Struct. Control Health
Monitoring, 11:223-237.
Zames, G. 1981. Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Transactions on Automatic Control 26: 301-320.