Fundamentals
Overview
What have we seen so far?
Properties of the response of dynamic systems and how these
are visible in the time and frequency domain
The FRF H(ω) and the TF H(s) for a SDOF system
What is shown in this class?
Continuous State-space Formulation
Multiple dof Systems
Solution of mdof Systems
Continuous versus Discrete Formulations
Institute of Structural Engineering Identification Methods for Structural Systems 2
Fundamentals
Overview
What have we seen so far?
Properties of the response of dynamic systems and how these
are visible in the time and frequency domain
The FRF H(ω) and the TF H(s) for a SDOF system
What is shown in this class?
Continuous State-space Formulation
Multiple dof Systems
Solution of mdof Systems
Continuous versus Discrete Formulations
Institute of Structural Engineering Identification Methods for Structural Systems 2
Fundamentals
Overview
What have we seen so far?
Properties of the response of dynamic systems and how these
are visible in the time and frequency domain
The FRF H(ω) and the TF H(s) for a SDOF system
What is shown in this class?
Continuous State-space Formulation
Multiple dof Systems
Solution of mdof Systems
Continuous versus Discrete Formulations
Institute of Structural Engineering Identification Methods for Structural Systems 2
Fundamentals
Overview
What have we seen so far?
Properties of the response of dynamic systems and how these
are visible in the time and frequency domain
The FRF H(ω) and the TF H(s) for a SDOF system
What is shown in this class?
Continuous State-space Formulation
Multiple dof Systems
Solution of mdof Systems
Continuous versus Discrete Formulations
Institute of Structural Engineering Identification Methods for Structural Systems 2
State Space Formulation
In the previous lecture, we discussed the usefulness of a system’s transfer,
or frequency response, function. However, as systems become more
complex, e.g. when their dimension increases, representing them with
differential equations or transfer functions becomes cumbersome. This
issue is more critical for multiple degree of freedom systems, where multiple
inputs and outputs may exist. To tackle this problem, we will introduce
here the continuous state-space form of a system, which will degenerate the
general nth order differential equation, describing the system dynamics,
with a first order matrix differential equation.
The continuous state-space form is described as:
¨x = Ax + Bu (state Eqn)
y = Cx + Du (observation Eqn)
for more info, see here
Institute of Structural Engineering Identification Methods for Structural Systems 3
State Space Formulation
Continuous State-space form
¨x = Ax + Bu (state Eqn)
y = Cx + Du (observation Eqn)
where
x is the state vector ∈ R
n
, it is a function of time
A is the state matrix ∈ R
nxn
, a constant
B is the input matrix ∈ R
nxp
, a constant
u is the input ∈ R
p
, a function of time
C, is the output matrix ∈ R
nxm
, a constant
D is the direct transition (or feedthrough) matrix ∈ R
mxp
, a constant
y is the output r ∈ R
m
, a function of time and it represents the quantity we
observe or measure
for more info, see here
Institute of Structural Engineering Identification Methods for Structural Systems 4
State Space Formulation
From Differential Equations to Continuous State-space form
There exist more than one methods for bringing a differential equation into
a continuous state space form. We will here describe the controllable
canonical form.
We will use a simple illustrative example to describe the process.
Assume a single degree of freedom (sdof) Single Input Single Output
(SISO) dynamical system described by the following equation of motion in
differential form:
d
3
y
dt
3
+ 2
d
2
y
dt
2
+ 3
dy
dt
+ 4y = 5u
Institute of Structural Engineering Identification Methods for Structural Systems 5
State Space Formulation
From Differential Equations to Continuous State-space form
d
3
y
dt
3
+ 2
d
2
y
dt
2
+ 3
dy
dt
+ 4y = 5u
Step 1:
Define the state-space vector. We define new variables, the so-called
system states. The state variables are the smallest possible subset of
system variables that can represent the entire state of the system at any
given time. The number of state variables n, is typically chosen equal to
the order of the system’s differential equation. In this case we have a 3rd
order equation, hence we define 3 states, corresponding to the derivatives
of the original system response variable y :
x
1
= y , x
2
=
dy
dt
=
dx
1
dt
, x
3
=
d
2
y
dt
2
=
dx
2
dt
x = [x
1
x
2
x
3
]
T
Institute of Structural Engineering Identification Methods for Structural Systems 6
State Space Formulation
From Differential Equations to Continuous State-space form
d
3
y
dt
3
+ 2
d
2
y
dt
2
+ 3
dy
dt
+ 4y = 5u (1)
Step 2:
Rewrite Equation (1), using the state variables:
dx
3
dt
+ 2x
3
+ 3x
2
+ 4x
1
= 5u (2)
Notice how the above expression contains 3 variables, but is a first order
differential equation.
Institute of Structural Engineering Identification Methods for Structural Systems 7
State Space Formulation
From Differential Equations to Continuous State-space form
Step 3:
Rewrite Equation (2) in matrix form:
˙x =
d
dt
x
1
x
2
x
3
=
0 1 0
0 0 1
−4 −3 −2
| {z }
A
x
1
x
2
x
3
+
0
0
5
| {z }
B
u
This equation is of the form ˙x = Ax + Bu
Notice how the last row of the state matrix A occurs by solving equation
(2) in terms of
dx
3
dt
.
Institute of Structural Engineering Identification Methods for Structural Systems 8
State Space Formulation
From Differential Equations to Continuous State-space form
Step 4:
Write the observation Equation, which delivers the actually desired
response quantity, i.e., y :
y =
1 0 0
| {z }
C
x
1
x
2
x
3
+
0
|{z}
D
u
This equation is of the form y = Cx + Du.
Institute of Structural Engineering Identification Methods for Structural Systems 9
State Space Formulation
C Tackling derivatives of the input u
The standard state space form, admits no derivatives of the input (load) u
on the right hand side. However, the original equation of motion could
include such derivatives, as for instance the case of a base-excited structure
in absolute coordinates (see Lecture 3, slide 11).
For instance, assume the same SISO dynamical system described by the
following equation of motion in differential form:
d
3
y
dt
3
+ 2
d
2
y
dt
2
+ 3
dy
dt
+ 4y = 2
d
2
u
dt
2
+ 5u
In order to bring this system into a canonical state-space form, we use a
transformation to a new variable ¯y:
y = 2
d
2
¯y
dt
2
+ 5¯y
Institute of Structural Engineering Identification Methods for Structural Systems 10
State Space Formulation
Tackling derivatives of the input
By substituting Equation (4) into Equation (3) we obtain:
2
d
2
dt
2
d
3
¯y
dt
3
+ 2
d
2
¯y
dt
2
+ 3
d ¯y
dt
+ 4¯y
+ 5
d
3
¯y
dt
3
+ 2
d
2
¯y
dt
2
+ 3
d ¯y
dt
+ 4¯y
=
2
d
2
u
dt
2
+ 5u
which implies that ¯y satisfies:
d
3
¯y
dt
3
+ 2
d
2
¯y
dt
2
+ 3
d ¯y
dt
+ 4¯y = u
This is now in regular form, and we may continue with the previous steps
for deriving the state-space form.
Institute of Structural Engineering Identification Methods for Structural Systems 11
Multiple DOF Systems
1
m
11
x
k
11
cx
( )
1
tF
2
m
( )
22 1
xxk
−
( )
22 1
xx
c
−
( )
2
tF
( )
22 1
xxk −
( )
22 1
xxc
−
FBD
(Lumped Mass System)
1
m
1
k
1
c
( )
1
tx
( )
1
tF
2
m
2
k
1
c
( )
2
tx
( )
2
tF
The equations of motion can be written as
m
1
¨x
1
+ (c
1
+ c
2
) ˙x
1
− c
2
˙x
2
+ (k
1
+ k
2
)x
1
− k
2
x
2
= F
1
(t)
m
2
¨x
2
+ c
2
˙x
2
− c
2
˙x
1
+ k
2
x
2
− k
2
x
1
= F
2
(t)
The system can be written in matrix form as follows:
m
1
0
0 m
2
¨x
1
¨x
2
+
c
1
+ c
2
−c
2
−c
2
c
2
˙x
1
˙x
2
+
k
1
+ k
2
−k
2
−k
2
k
2
x
1
x
2
=
F
1
(t)
F
2
(t)
Eq. (1)
Institute of Structural Engineering Identification Methods for Structural Systems 12
Multiple DOF Systems
From sdof to modf
We have so far seen the general steps for converting a sdof system
described by an ordinary differential equation into state-space form.
We will now demonstrate how the same context can be applied into
multiple degree of freedom systems (mdof).
We will use an approximation of a structural system as the main example
for our demonstration (see next slide)
Institute of Structural Engineering Identification Methods for Structural Systems 13
State Space Equation Formulation for MDOF systems
2dof Mass Spring System
or otherwise more compactly, in matrix form:
M ¨x
d
+ C ˙x
d
+ Kx
d
= u
where x
d
=
x
1
x
2
T
and u =
F
1
F
2
T
We now introduce the augmented state vector:
x =
x
d
˙
x
d
T
=
x
1
x
2
˙x
1
˙x
2
T
. Then,
˙x =
˙x
1
˙x
2
¨x
1
¨x
2
=
0 0 1 0
0 0 0 1
−M
−1
K
−M
−1
C
x
1
x
2
˙x
1
˙x
2
+
0 0
0 0
M
−1
F
1
F
2
Institute of Structural Engineering Identification Methods for Structural Systems 14
State Space Equation Formulation for MDOF systems
State Equation
This is rewritten as:
˙x =
O
2x2
I
2x2
−M
−1
K
2x2
−M
−1
C
2x2
x +
0
2x2
M
−1
u
We therefore obtain the following equivalent state-space formulation:
⇒ ˙x = Ax + Bu
where it is reminded that x
d
=
x
1
x
2
T
, u =
F
1
F
2
T
For a general n-dimensional system (n dofs), matrices A and B are
obtained as:
A =
O
nxn
I
nxn
−M
−1
K
nxn
−M
−1
C
nxn
B =
0
nxn
M
−1
Institute of Structural Engineering Identification Methods for Structural Systems 15
State Space Equation Formulation
Observation Equation
The observation equation contains the quantities we “observe”, i.e.
measure, using relevant sensors. Assume we monitor (measure)
both displacements x
1
, x
2
, e.g. via a laser sensor. Then the
“observation vector” is:
y =
1 0 0 0
0 1 0 0
| {z }
C
x
1
x
2
˙x
1
˙x
2
+
O
2×2
u(t)
Assume we monitor (measure) the 2nd dof velocity, ˙x
2
, e.g. via a
geophone sensor. Then the “observation vector” is:
y =
0 0 1 0
| {z }
C
x
1
x
2
˙x
1
˙x
2
+
O
1×2
u(t)
Institute of Structural Engineering Identification Methods for Structural Systems 16
State Space Equation Formulation
Observation Equation
Assume we monitor both accelerations, ˙x
1
, ˙x
2
via accelerometers:
y =
−M
−1
K
2x2
−M
−1
C
2x2
| {z }
C
x
1
x
2
˙x
1
˙x
2
+
M
−1
2x2
| {z }
D
u
If we only monitor the 1st dof acceleration, ˙x
1
, the “observation vector”
is:
y =
−(k
1
+ k
2
)/m
1
k
2
/m
1
−(c
1
+ c
2
)/m
1
c
2
/m
1
| {z }
C
x
1
x
2
˙x
1
˙x
2
+ 1/m
1
|{z}
D
F
1
The observation vector is written in matrix form as y = Cx + Du
Institute of Structural Engineering Identification Methods for Structural Systems 17
State Space Equation Formulation
Note
Using the state space representation we have converted a 2nd order
ODE into an equivalent 1st order ODE system.
We can now use any of the aforementioned 1st order ODE
integration methods in order to convert the continuous system into
a discrete one and obtain an approximate solution
For instance MATLAB’s ode45, which is a Runge Kutta integration
scheme may be used.
What are other integration schemes that may be utilized?
Institute of Structural Engineering Identification Methods for Structural Systems 18
Numerical Integration for 1st order ODEs
Using these methods a continuous system is brought into an equivalent
discrete formulation and an approximative solution is sought. 1st order
ODE Integration Methods
Assume
dy
dt
= f (t, y(t)), y(t
0
) = 0
Forward Euler Method
y
n+1
= y
n
+ hf (t
n
, y
n
)
where h is the integration time step. This explicit expression is
obtained from the truncated Taylor Expansion of y (t
n
+ h). More info
here
Backward Euler Method
y
n+1
= y
n
+ hf (t
n+1,
y
n+1
)
This implicit expression (since y
n+1
is on the right hand side) is
obtained from the truncated Taylor Expansion of y(t
n+1
− h). where h
is the integration time step. This explicit expression is obtained from
the truncated Taylor Expansion of y (t
n
+ h).
Institute of Structural Engineering Identification Methods for Structural Systems 19
Numerical Integration for 1st order ODEs
2nd Order Runge Kutta (RK2)
k
1
= hf (t
n
, y
n
), k
2
= hf (t
n
+
1
2
h, y
n
+
1
2
k
1
)
y
n+1
= y
n
+ k
2
+ O(h
3
)
More info here
4th Order Runge Kutta (RK4) - MATLAB ode45
funcction
k
1
= hf (t
n
, y
n
), k
2
= hf (t
n
+
1
2
h, y
n
+
1
2
k
1
)
k
3
= hf (t
n
+
1
2
h, y
n
+
1
2
k
2
), k
4
= hf (t
n
+ h, y
n
+ k
3
)
y
n+1
= y
n
+
1
6
k
1
+
1
3
k
2
+
1
3
k
3
+
1
6
k
4
+ O(h
5
)
More info here
Institute of Structural Engineering Identification Methods for Structural Systems 20
Solution of mdof systems
Laplace Transform for MDOF Systems
Assume the previous 2-DOF system, which we bring to a state-space form,
i.e., the system may be summarized by the following 1st order ODE as
follows:
˙x(t) = Ax(t) + Bu(t) state-space or process equation
y(t) = Cx(t) + Du(t) measurement or observation equation
Since we are now dealing with higher dimensions, let us now define the
Laplace Transform for a vector of for instance 4 components:
L
x
1
(t)
x
2
(t)
x
3
(t)
x
4
(t)
=
L{x
1
(t)}
L{x
2
(t)}
L{x
3
(t)}
L{x
4
(t)}
= X (s)
Also, the Laplace property in relation to differentiation:
L{ ˙x(t)} = sX (s) − X (0) applies now in vector form.
Institute of Structural Engineering Identification Methods for Structural Systems 21
Solution of mdof systems
Laplace Transform for MDOF Systems
Let us therefore return to the original state-space formulation and apply the
Laplace Transform therein:
˙x(t) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
⇒
sX (s) − X (0) = AX (s) + BU(s)
Y (s) = CX (s) + DU(s)
(sI − A)X (s) = BU(s) + X (0) ⇒
X (s) = (sI − A)
−1
BU(s) + (sI − A)
−1
X (0)
Assuming D = 0 and 0 Initial Conditions the observation equation yields
X (s) = (sI − A)
−1
BU(s)
Institute of Structural Engineering Identification Methods for Structural Systems 22
Solution of mdof systems
Laplace Transform for MDOF Systems
The Input-Response Transfer Function
The previous relationships provides the connection, in the form of a transfer
function, H(s), between the input (load), U(s), and the response (output),
X (s):
H(s) = (sI − A)
−1
B
H(s) can reveal significant information about the system as we have
already seen for the case of SDOF systems.
However, we are interested in obtaining the solution in a time domain form,
x(t). In doing so, we may use the Inverse Laplace transform:
x(t) = L
−1
(sI − A)
−1
BU(s)
Institute of Structural Engineering Identification Methods for Structural Systems 23
Solution of mdof systems
Reminder: Impulse Response Function
The IRF is provided through the Inverse Laplace of the FRF (from
frequency to time domain):
h(t) = L
−1
H(ω)
The Input-Measurement Transfer Function
Of additional interest is the connection, in the form of a transfer
function, H
Y
(s), between the input (load), U(s), and the
measurement vector Y (s)
This may be obtained as:
Y (s) = {C(sI − A)
−1
B}U(s) ⇒
H
F
(s) = C(sI − A)
−1
B
Institute of Structural Engineering Identification Methods for Structural Systems 24
Solution of mdof systems
Note: In calculating the TF the inverse of a matrix needs to be calculated:
by definition: (sI − A)
−1
=
adj(sI − A)
det(sI − A)
where adj(Z) =
h
ij
|Z
ij
i
, Z
ij
: det(Z) without row i, column j
Then, det(sI − A)Y (s) = C [adj(sI − A)B] U(s)
where det(sI − A) = s
n
+ a
n−1
s
n−1
+ · · · + a
0
is the characteristic
polynomial with roots λ
1
, · · · , λ
n
These roots are directly related to the eigenvalues of the original eigenvalue
problem: (K − Mω
2
)φ = 0
Institute of Structural Engineering Identification Methods for Structural Systems 25
Solution of mdof systems
mdof Structural Systems
As an alternative, we can obtain the Transfer Function (TF) for the
MDOF, directly from the Laplace Transform of the dynamic
equation of motion (in matrix form):
m¨x + c ˙x + kx = f ⇒
ms
2
+ cs + k
X(s) = F(s) ⇒
TF: H(s) =
ms
2
+ cs + k
−1
For underdamped systems with symmetric matrices like the above,
using partial fraction expansion the TF can be written as:
H(s) =
m
X
k=1
A
k
s − λ
k
+
A
∗
k
s − λ
∗
k
* denotes complex conjugate
where
λ
k
= −ζ
k
ω
k
+ iω
k
q
1 − ζ
2
k
Institute of Structural Engineering Identification Methods for Structural Systems 26
Solution of mdof systems
For s = iω the TF is evaluated on the Imaginary axis yielding the
Frequency Response Function (FRF) with the following terms
H
ij
(ω) =
m
X
k=1
k
A
ij
iω − λ
k
+
k
A
∗
ijk
iω − λ
∗
k
* denotes complex conjugate
The term H
ij
(ω) corresponds to a particular output response at point i due
to an input force at point j. This follows from the property L {δ(t)} = 1.
Since for a structural system m, c, k are symmetric we have that H
ij
= H
ji
(reciprocity). This means that each term could be experimentally evaluated
by applying impact at a point i and recording the response at point j.
The terms α
ijk
are named the residues and they are related to the mode
shape (eigenvector) matrix Φ as:
k
A
ij
= q
k
φ
ik
φ
jk
Institute of Structural Engineering Identification Methods for Structural Systems 27
Solution of mdof systems
Solving MDOF systems via the continuous state-space form:
Other than the inverse Laplace transform, we may solve mdof
systems by using the state-space formulation:
Examine the state-space system with no input, u = 0:
˙x = Ax given I.C. at x(0)
Assume the above system has a solution of the type x = Ce
st
φ,
where φ is a vector.
Plug this in the differential equation to obtain:
se
st
φ = Aφe
st
⇒ [sI − A] φ = 0
Institute of Structural Engineering Identification Methods for Structural Systems 28
Solution of mdof systems
This is a standard eigenproblem with eigenvalues s obtained from
det(sI − A) = 0 and φ are the corresponding eigenvectors
Then, the total solution of the system is obtained through
superposition as
x(t) = C
1
φ
1
e
s
1
t
+ C
2
φ
2
e
s
2
t
+ · · · + C
n
φ
n
e
s
n
t
Hence, at t = 0 ⇒ x(0) = C
1
φ
1
+ C
2
φ
2
+ · · · + C
n
φ
n
=
=
φ
1
φ
2
· · · φ
n
C
1
C
2
.
.
.
C
n
⇒
C
1
C
2
.
.
.
C
n
= Φ
−1
x(0)
where Φ is the matrix of eigenvectors, which is directly related to the
structure’s modal shape matrix.
Institute of Structural Engineering Identification Methods for Structural Systems 29
Solution of mdof systems
We define the State Transition Matrix M(t) as
M(t) = Φ
e
s
1
t
· · · 0
.
.
.
.
.
.
.
.
.
0 · · · e
s
n
t
Φ
−1
and x(t) = M(t)x(0)
If Initial Conditions are specified at a generic point t
0
we get:
M(t − t
0
) = Φ
e
s
1
(t−t
0
)
· · · 0
.
.
.
.
.
.
.
.
.
0 · · · e
s
n
(t−t
0
)
Φ
−1
and x(t) = M(t − t
0
)x(t
0
)
Institute of Structural Engineering Identification Methods for Structural Systems 31
Multiple dof Systems
State Transition Matrix Properties
A. Assume we would like to obtain the connection of the response between
three successive time interval, t
0
,t
1
,t
2
.
x(t
1
) = M(t
1
− t
0
)x(t
0
)
x(t
2
) = M(t
2
− t
1
)x(t
1
)
⇒
x(t
2
) = M(t
2
− t
1
)M(t
1
− t
0
)x(t
0
)
Which by definition means that
M(t
2
− t
0
) = M(t
2
− t
1
)M(t
1
− t
0
)
B. Now consider t
2
= t
0
(so we move 1 step back in time) then
I = M(t
0
− t
1
)M(t
1
− t
0
) ⇒
M(t
1
− t
0
)
−1
= M(t
0
− t
1
)
Comparing the above to the properties of an exponential we observe:
e
α(t
2
−t
1
)
e
α(t
1
−t
0
)
= e
α(t
2
−t
o
)
(e
α(t
1
−t
0
)
)
−1
= e
α(t
0
−t
1
)
⇒ similar traits
Institute of Structural Engineering Identification Methods for Structural Systems 32
Multiple dof Systems
The Matrix Exponential
Knowing that M(t) behaves like an exponential, let us assume A is
an m × n square matrix and
define: e
At
= I + At +
1
2!
A
2
t
2
+
1
3!
A
3
t
3
· · ·
If we plug this in ˙x = Ax, where A is the state-space matrix:
A +
2
2!
A
2
t +
3
3!
A
3
t
2
· · · = A(I + At +
1
2!
A
2
t
2
+
1
3!
A
3
t
3
· · · )
we notice that e
At
is a solution to our original problem! Indeed, we
may write the solution as:
˙x = Ax ⇒
x(t) = e
At
x(0), for given x(0)
x(t) = e
A(t−t
0
)
x(t
0
), for given x(t
0
)
Institute of Structural Engineering Identification Methods for Structural Systems 33
Multiple dof Systems
The Matrix Exponential
This implies that the state transition matrix actually is:
M(t) = e
At
In fact, the full solution to ˙x = Ax + Bu is proved to be (for IC x(0)):
x(t) = e
At
x(0) +
Z
t
0
e
A(t−τ )
Bu(τ)dτ
Notice the resemblance to Duhamel’s Integral
It is interesting to therefore note the equivalence between SDOF and
MDOF systems.
Institute of Structural Engineering Identification Methods for Structural Systems 34
Multiple dof Systems
from Continuous to Discrete time
The continuous state space form was derived as:
¨x(t) = Ax(t) + Bu(t) (continuous state Eqn)
y(t) = Cx(t) + Du(t) (observation Eqn)
Information from sensors (monitoring) comes at discrete time intervals,
with measurements obtained usually at a fixed interval, e.g. every T
seconds. For such systems, it is more convenient to use a discretized
representation (state-space) to describe the system:
x((k + 1)T ) = G(T )x(KT ) + H(T )u(KT ) (discrete state Eqn)
y(kT ) = Cx(kT ) + Du(KT ) (observation Eqn)
where k is an integer. The discretized state-space matrices G, H depend on
the value of the sampling interval T , while the observation equation
remains the same in both the continuous and discrete form (since it is
simply a linear equation, and not a differential equation).
Institute of Structural Engineering Identification Methods for Structural Systems 35
Multiple dof Systems
from Continuous to Discrete time
The discrete matrices may be obtained by using the solution of the
continuous state space equation:
x((k + 1)T ) = e
A(k+1)T
x(0) + e
A(k+1)T
Z
(k+1)T
0
e
−Aτ
Bu(τ)dτ
x(kT ) = e
AkT
x(0) + e
AkT
Z
kT
0
e
A(−τ )
Bu(τ)dτ
Using the above equations, we express x((k + 1)T ) as a function of x(kT ):
x((k + 1)T ) = e
AT
x(kT ) + e
A(k+1)T
Z
(k+1)T
kT
e
−Aτ
Bu(τ)dτ
Institute of Structural Engineering Identification Methods for Structural Systems 36
Multiple dof Systems
from Continuous to Discrete time
By assuming u(t) = u(kT ) within the interval [kT (k + 1)T ] (zero-order
hold assumption), we may take this expression together with matrix B, out
of the integral to obtain:
x((k + 1)T ) = e
AT
x(kT ) + e
A(k+1)T
Z
(k+1)T
kT
e
−Aτ
dτ Bu(kT )
τ ∈ [kT , (k + 1)T ]
With an appropriate change of variables, it is proven that the discrete
state-space matrices are obtained as:
G(T ) = e
AT
H(T ) = A
−1
e
AT
− I
B
for more info, see here
Institute of Structural Engineering Identification Methods for Structural Systems 37
