October 2001

NASA/TM-2001-211229

System Identification of a Vortex Lattice

Aerodynamic Model

Jer-Nan Juang

Langley Research Center, Hampton, Virginia

Denis Kholodar and Earl H. Dowell

Duke University, Durham, North Carolina
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October 2001

NASA/TM-2001-211229

System Identification of a Vortex Lattice

Aerodynamic Model

Jer-Nan Juang

Langley Research Center, Hampton, Virginia

Denis Kholodar and Earl H. Dowell

Duke University, Durham, North Carolina
Available from:

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)

7121 Standard Drive 5285 Port Royal Road

Hanover, MD 21076-1320 Springfield, VA 22161-2171

(301) 621-0390 (703) 605-6000
System Identification of a 

Vortex Lattice Aerodynamic Model

Jer-Nan Juang1, Denis Kholodar 2 and Earl H. Dowell 3

NASA Langley Research Center, Hampton, Virginia 23681-0001

Duke University, Durham, North Carolina 27708-0300

Abstract

The state-space representation of an aerodynamic vortex lattice model is considered from 

a classical and system identification perspective. Using an aerodynamic vortex model as

a numerical simulator of a wing tunnel experiment, both full state and limited state data

or measurements are considered. Two possible approaches for system identification are

presented and modal controllability and observability are also considered. The theory

then is applied to the system identification of a flow over an aerodynamic delta wing and 

typical results are presented.

---------------------------

1

Principal Scientist, Structural Dynamic Branch. Fellow AIAA

2

Research Assistant, Department of Mechanical Engineering and Material Science.

3

J. A. Jones Professor, Department of Mechanical Engineering and Material Science and Dean Emeritus, 

Pratt School of Engineering. Fellow AIAA. Director, Center for Nonlinear and Complex Systems.
2

Nomenclature

Γ vortex strength

A state matrix

B input influence matrix

C output influence matrix

D direct transmission matrix

U input matrix

li

discrete-time eigenvalues

Λ diagonal matrix of eigenvalues

ψi

eigenvectors corresponding to eigenvalues, λi

Ψ matrix of eigenvectors corresponding to eigenvalue matrix, Λ

Yi Markov parameters

H(i) Hankel matrices

X pcontrollability matrix

Op observability matrix

w downwash velocity

δ percent error

bm

controllability vector
3

Introduction

In recent years, several studies have been conducted at Duke University of an

aeroelastic delta wing system. In particular, reduced order unsteady aerodynamic models

have been used to predict the flutter, limit cycle oscillations (LCO) and gust forced

response of such a wing. For the case of incompressible, inviscid and irrotational flow, an 

unsteady vortex lattice method was used for time domain analyses. Moreover, once such

a model was created, it was used to create a reduced order model (ROM) which allowed a 

reduction in the order of the vortex lattice code from a thousand degrees of freedom or

more to the order of ten degrees of freedom while retaining essentially the same accuracy 

for the representation of fluid forces acting on a wing. 

The development of such a vortex lattice model and a reduced order aerodynamic 

model based upon aerodynamic eigenmodes can be found in Ref. [1]. However, the

representation of unsteady aerodynamic flow fields in terms of global aerodynamic

modes can be developed in a variety of ways. In Ref. [2], the authors used Proper

Orthogonal Decomposition modes and also a system identification model for a delta wing 

to obtain a reduced order model. The current work studies the system identification of a

vortex lattice model in greater depth with a view to developing a methodology that can be 

used with wind tunnel experimental data. The vortex lattice model is used here as a test

bed for a numerical experiment in a continuation of the work begun in Ref. [2].

Delta Wing Vortex Model

The flow about a cantilevered half-span delta wing is assumed to be

incompressible, inviscid and irrotational. An unsteady vortex lattice method can be used

to model this flow. A typical planar vortex lattice mesh for the three-dimensional flow

over a wing is shown in Figure 1. The delta wing and the wake are divided into a number 

of vortex elements. Point vortices are placed on the wing and in the wake at the quarter

chord of the elements. At the three-quarter chord of each plate element a collocation

point is placed for the downwash and the velocity induced by the discrete vortices is set

equal to the downwash (fluid vertical velocity on the wing) imposed by the prescribed
4

delta wing motion or the gust field. Unsteady vorticity is shed into the wake and

convected with the freestream velocity. The equations expressing these relationships are

well described in Ref [1].

State-Space Representation

The relationship between a Vortex Lattice (VL) model and a state-space model 

developed by system identification methods will be established in this section. For 

simplicity, we begin with the assumption that full state information is given. This allows 

us to develop a formulation to gain basic insight before pursuing further development of 

the case where only partial state data are available. 

Full-State Data or Measurement

The VL model can be described by1

AΓ + (k 1) + BΓ(k) = + Du k( 1) % % %

 (1)

where Γ + (k 1) is the strength vector of the vortex at the time step k+1, A%

 and B% are

aerodynamic coefficient matrices, and D% is a transfer matrix for determining the

relationship between the global vortex lattice mesh and the local vortex lattice mesh on

the delta wing itself, and u is the downwash vector. Expressions for A%, B% and D% are 

given in Ref. [1]. Assume that there is a state-space model in the form

( 1) ( ) ( )

( ) ( ) ( )

x k Ax k Bu k

k Cx k Du k

+ = +

Γ = +

 (2)

where A is the state matrix, B is the input influence matrix, C is the output influence

matrix, and D is the direct transmission matrix. Equation (2) gives the same map

(relationship) as shown in Eq. (1) from u to Γ . Is this assumption valid? 

To answer this question, let us proceed as follows. Assuming that all states are

measurable, the output matrix C in Eq. (2) is square and invertible. Rewrite the bottom

equation of Eq. (2) to become
5

1 1 x(k) C (k) C Du k( ) − −

= Γ − (3)

Substituting Eq. (3) into the top equation of Eq. (2) yields

1 1 1 1 C (k 1) AC k( ) [B AC D u] (k) C Du k( 1) − − − − Γ + = Γ + − + + (4)

Premultiplying Eq. (4) by C produces

1 1 (k 1) CAC (k) [CB CAC D u] (k) Du k( 1) − − Γ + = Γ + − + + (5)

Comparison of Eqs. (1) and (5) results in the following equalities

1 1

1

1 1 1

CAC A B

D A D

CB CAC D A BA D

− −

−

− − −

= −

=

= = −

% %

% %

% % % %

 (6)

From Eq. (6), it is clear that the state-space model is not unique in the sense that we have 

the freedom to choose C. Regardless of the choice of C, however, the eigenvalues of the 

VL model, i.e., eigenvalues of 1 A B−−

% %

, are identical to those of A (state matrix), as long

as C is invertible. For the case where C is an identity matrix, i.e., C I = , Eq. (6) becomes

1

1

1 1

A A B

D A D

B AD A BA D

−

−

− −

= −

=

= = −

% %

% %

% % % %

 (7)

Thus, we have shown that the VL model, Eq. (1), has a state-space representation, Eq.

(2), with system matrices, A, B, and D, uniquely determined by Eq. (7), assuming that the 

output matrix C is an identity matrix. This state-space model may be used for model 

reduction and system identification.

Assume that we are given a (discrete time) sequence of 

Γ = [Γ(1) Γ Γ (2) L l( )] (8)

and

U = [u(1) u u (2) L l( )] (9)

This may be a sequence of experimental data obtained from measurements or, as in the

present case, numerical data from the vortex lattice model. One may use these sequences 

Γ and U to identify a set of system matrices, A, B, C, and D. If the identified C is an
6

invertible square matrix, the eigenvalues of the state matrix A are, indeed, the eigenvalues 

of the VL model, i.e., the eigenvalues of 1 A B−

−

% %

. To make C square and invertible

requires that the number of measurements (rows) is identical to the number of states

(columns). Furthermore, all rows of U must be linearly independent for the system

identification to be valid. If the rows of U are not linearly independent, the matrix B

cannot be properly identified. Does this mean that the eigenvalues of A may not be

identified properly? The answer is "no". The eigenvalues of A can be fully identified if 

some of the rows can sufficiently excite the system eigenvalues. See, for example, Ref.

[3].

Limited State Data or Measurements

For simplicity, we have first developed the basic formulation based on a set of full 

state data or measurements. In practice, the spacial or temporal measurements may be

extremely limited probably tens of spacial locations and/or time steps. For example, what 

if one is given only the first few elements of Γ (k) for k = 1,2, , L l generated by Eq. (1)? 

Can we use this very limited number of elements to identify the eigenvalues of 1 A B−−

% %

? 

One might intuitively conclude that it is very difficult, if not impossible. But, in fact,

some progress can be mode as follows.

Let Eq. (1) be transformed to a new form in terms of modal coordinates, i.e.,

( 1) ( ) ( 1) m m m Γ k + + ΛΓ k = + Duk %

 (10)

where

[ ]

1 1

1 2

1

1 1

diag

( ) ( )

n

m

m

A B

k k

D A D

λ λ λ − −

−

− −

Λ = Ψ   Ψ =  

Γ = Ψ Γ

= Ψ

% % L

% % %

The quantities λ1 2 λ λ L n

are the eigenvalues of the VL model,

andψ1 2 ψ ψ L n are the corresponding eigenvectors that form the matrix Ψ. 

Assume that we have only one input, i.e., u(k+1) in Eq. (10) is a scalar. All the modes

will be excited if the column vector Dm

%

 has no zero elements in it, and u(k+1) is
7

nonzero. If some elements of Dm

%

 are zero, the corresponding eigenvectors will not be

excited. Other inputs may be added to excite such modes, of course. The elements of Dm

%

with relatively larger magnitude will excite the corresponding eigenmode to have a larger 

response. Assume that we have excited all the modes by properly choosing some input

locations and their excitation signal, i.e., none of the columns in

Γm = [Γm(1) Γ Γ m m (2) L l( )]

is a zero vector. The output data or measurements will become

Γ = [Γ(1) Γ(2) L Γ(l)] = Ψ[Γm

(1) Γ Γ m m (2) L l( )]

If Ψ is invertible, each row of Γ should include all information of the VL eigenvalues

and eigenvectors. In theory, one should be able to extract the information of eigenvalues

from one row of Γ unless there exists some repeated eigenvalues. 

Let us rewrite Eq. (1) to become

ˆ ˆ

( 1) ( ) ( 1)

ˆ

( ) ( )

k A k Du k

y k C k

Γ + = Γ + +

= Γ

 (11)

where 

1

1

ˆ

ˆ

A A B

D A D

−

−

= −

=

% %

% %

and ˆ

C is an m n × matrix with m n < . Each row of ˆC could have a unity at one location

and zeros at all other locations so that y(k) includes only the desired elements in Γ( ) k . 

From Eq. (11), the following matrix equation can be easily derived, i.e.,

2

1

ˆ 0 0 0 0 ( ) ( )

ˆ ˆ ˆ ˆ ( 1) 0 0 0 ( 1)

( )

( 1) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( 1) ˆ ˆ 0

p

p

y k C u k

y k CA CD u k

k

y k p u k p

CA CA D CAD CD −

−

                + +       =  Γ +                + −    + −        

M M M O M M M

 (12)

or

( ) ( ) ( ) p p p p y k = Θ Γ k + Υ u k (13)

where
8

2

1

( ) ( )

( 1) ( 1)

( ) ; ( ) ;

( 1) ( 1)

ˆ 0 0 0 0

ˆ ˆ ˆ ˆ 0 0 0

;

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0

p p

p p

p

p

y k u k

y k u k

y k u k

y k p u k p

C

CA CD

CA CA D CAD CD −

−

   

    + +

= =    

   

     + −    + −

         

Θ =   Υ =  

 

     

M M

M M O M M

where p is an integer that must be chosen to makeΘp a square matrix of pm n × with

pm=n and m is the number of outputs. There is a great chance that such an integer p does 

not exist. We will discuss this case later. The quantity Υp

 is a pm × pr matrix with r

being the number of inputs.

The quantity ( ) py k is a pm ×1column vector stacking together the m ×1 vectors, 

y(k), y(k+1), ..., y(k+p-1) from different time steps, where m is the number of outputs. 

Similarly, the quantity ( ) u k p is a pr ×1column vector consisting of r ×1 vectors, u(k), 

u(k+1), ..., u(k+p-1), where r is the number of outputs. Solving Eq. (13) for Γ( ) k yields

1 1 ( ) ( ) ( ) p p p p p k y k u k − − Γ = Θ − Θ Υ (14)

Substituting Γ( ) k and Γ + (k 1) from Eq. (14) into Eq. (11) and pre-multiplying the

resultant equation by Θp

 produces

ˆ ˆ 1 1 ˆ

( 1) ( ) ( ) ( 1) ( 1) p p p p p p p p p p p y k A y k A u k u k Du k − − + = Θ Θ − Θ Θ Υ + Υ + + Θ + (15)

Assume that the number of inputs is r, i.e., u(k+1) is a r ×1vector. Define ˆ

Υp as

p

ˆ ˆ

 with the first zero columns replaced by Υp p = Υ Θ γ D (16)

Equation (15) becomes

ˆ 1 1 ˆ ˆ ( 1) ( ) ( ) ( 1) p p p p p p p p p p y k A y k A u k u k − − + = Θ Θ − Θ Θ Υ + Υ + (17)

For easy reference, let us call Eq. (17) the Generalized Vortex Lattice (GVL) model. This 

equation has a state-space representation similar to Eq. (2), i.e.,
9

( 1) ( ) ( )

( ) ( ) ( )

p p

p p p p

x k Ax k B u k

y k C x k D u k

+ = +

= +

 (18)

where 

1

1

ˆ

ˆ ˆ [ ]

ˆ

p p

p p p p p

p pm pm

p p

A A

B A

C I

D

−

−

×

= Θ Θ

= Θ Θ Υ − Υ

=

= Υ

 (19)

The output matrix Cpis chosen to be an identity matrix. Equation (18) is a state-space 

model representing the map from the input vector ( ) u k p of pr ×1 to the output vector

( ) py k of pm ×1. Now what we are looking for is the map from input vector u k( ) of r ×1

to the output vector y k( ) of m ×1. Careful examination of ˆ

Υp as defined in Eq. (16)

reveals that 

p ( 1)

ˆ ˆ

0 p p D pm× −p r

  Υ − Υ=Θ    (20)

where 0pm× − ( p r 1) is an pm × − ( p r 1) zero matrix. Equation (18) thus reduces to 

( 1) ( ) ( )

( ) ( ) ( )

x k Ax k Bu k

y k Cx k Du k

+ = +

= +

 (21)

where 

1

( 1)

ˆ

ˆ

0

ˆ

First rows of 

p p

p

m m m p m

p

A A

B D

C I

D m D

−

× × −

= Θ Θ

= Θ

=    

= Θ

 (22)

The measurement or data output equation for y(k) is obtained by taking the first m rows

of ( ) p

y k shown in Eq. (18) and noting that pm pm I ×is an identity matrix and ˆΥp is a

special matrix, i.e., the upper right-hand entries are all zeros. The matrix A is related to ˆA
10

by a similarity transformation such that they share the same eigenvalues. We thus

conclude that the eigenvalues for the VL model can be obtained from the state-space 

model shown in Eq. (21). The fundamental assumption is that the matrix Θp

 must be

invertible. The matrixΘpis commonly referred to as the observability matrix in the

control field. If some of the eigenvalues of the VL model are not observable, then

Θp will not be full rank and thus not invertible. Then the identified state-space matrix A

shown in Eq. (21) would not include the un-observable eigenvalues. The maximum rank 

of Θp is the number of states, n. Therefore, even though we have an oversized pm n ×

matrixΘp with pm n > , a good system identification technique should be able to identify 

a minimum-size matrix A that would include only the observable and controllable VL

eigenvalues and eigenvectors.

System Identification

Several methods may be used for system identification (see Ref. [3]). Each has its 

own disadvantages and advantages. Here two simple approaches are presented, (1)

Generalized Vortex Lattice (GVL) model identification and (2) Eigensystem Realization

Algorithm (ERA).

GVL Model identification

The GVL model identification begins with Eq. (17). Let us rewrite Eq. (17) to

become

ˆ

( 1) ( ) ( ) ( 1) p p p p p p y k + = Ay k − AΥ u k + Υ + u k (23)

where the state matrix A is defined in Eq. (22). Let the time index k run from 1 to l . 

Equation (16) will then produce the following matrix equation.

ˆ

(2) (1) (1) (2) Yp = AYp − AΥpU U p + Υp p

or 
11

(1)

ˆ

(2) (1)

(2)

p

p p p p

p

Y

Y A A U

U

 

  

= − Υ Υ        

 (24)

where (1) Yp

, (2) Yp, (1) U p, and (2) U p are defined by

( ) ( ) ( 1) ( 1)

( ) ( 1) ( 1)

( 1) ( 2) ( )

( 1) ( ) ( 2)

Yp p p p k y k y k y p k

y k y k y p k

y k y k y p k

y k p y k p y k

=   + −+−  

  + −+−

  + + − +

=

  + − + + −

L l

L l

L l

M M O M

L l

and

( ) ( ) ( 1) ( 1)

( ) ( 1) ( 1)

( 1) ( 2) ( )

( 1) ( ) ( 2)

U p p p p k u k u k u p k

u k u k u p k

u k u k u p k

u k p u k p u k

=   + ++−  

  + −+−

  + + − +

=

  + − + + −

L l

L l

L l

M M O M

L l

with k = 1 and 2. Note that A is an n n × matrix, AΥp is an n × pr matrix, and ˆΥp is

also an n × pr matrix. Correspondingly, (1) Yp and (2) Yp are both pm p × − ( ) l matrices, 

and (1) U p

 and (2) U p are both pr p × − ( ) l matrices. 

In Eq. (24), we have three matrices, A, AΥp, and ˆΥp, with a total of n × + (n 2 ) pr

unknowns to be determined. The matrix

(1)

(1)

(2)

p

p

p

Y

U

U

 

 

 

   

 is of the size (n + 2 pr p ) × − ( ) l . If

the data length l is given such that l − p ≥ +n 2 pr , then the maximum rank of the

matrix is n + 2 pr if all the rows are linearly independent. Unfortunately, the matrices

(1) U p and (2) U p have all rows in common except the first row of (1) U p and the last row 

of (2) Up. This means that the three matrices, A, AΥp, and ˆΥp cannot be solved

uniquely. To solve the state matrix uniquely, let us define
12

ˆ ˆ 0 0 n× × r A p p n r

  Υ =   − Υ + Υ       (25)

Using Eq. (25), Eq. (24) can be rearranged to become

1

(1)

ˆ

(2)

(1)

p

p

p

Y

Y A

U +

    = Υ       

 (26)

The matrix 1(1) U p+ is the union of (1) U p and (2) U p, i.e., the (1) U p plus the last r row of 

(2) Up. Now, assume that an input signal is generated such that all rows of 1(1) U p+ are

linearly independent, and the resulting (1) Yp from the measurement also has linearly

independent rows. Equation (26) can then be solved to yield

†

1

(1)

ˆ

(2)

(1)

p

p p

p

Y

A Y

U +

    Υ =        

 (27)

where †means pseudo-inverse. This is a least-squares solution if

1

(1)

(1)

p

p

Y

U +

 

   

 has more

columns than rows. 

Equation (27) provides the state matrix A as shown in Eq. (22) which in turns

produces the eigenvalues of the VL model. Furthermore, Eq. (27) also produces the input 

matrix B shown in Eq. (22) that is the first r columns of ˆΥp [see Eqs. (16) and (22)]. 

Finally, the direct transmission term D is the first m rows of B [see Eq. (22)]. 

Equation (27) for system identification is straightforward and intuitive. 

Nevertheless, it does not work for the case where the data length is so short that

1

(1)

(1)

p

p

Y

U +

 

   

 has more rows than columns, i.e., more spacial measurement points than time

data steps. On the other hand, the state matrix A produced by Eq. (23) will, in general,

not be a minimum realization in the sense that it is oversized. In other words, the integer 

p that produces the row number of (1) Yp and 1

(1) U p+

 may be chosen larger than the

number of states.
13

Eigensystem Realization Algorithm

Another approach that produces a minimum-size model is based on the minimum

realization theory. This approach uses a sequence of responses generated by a pulse input 

to the real system and then the computation and inversion of the system transfer functions 

is obtained from the input and output data (see Chapter 5 and 6 of Ref. [3]). Let

(0) 1 ( 1, 2, , ) iu = =i r K and ( ) 0 ( 1, 2, ) i

u k k = = K be substituted into Eq. (21). When the

substitution is performed for each input element (0) 1 ( 1, 2, , ) i

u = =i r K , the results can

be assembled into a pulse response matrix Y with dimension m by r as follows:

1

0 1 2 , , , ,

k

Y D Y CB Y CAB Yk CA B−

= = = = L (28)

The constant matrices in the sequence are known as Markov parameters (See Ref. [3]).

System identification begins by forming the generalized Hankel matrix H(0), 

composed of the Markov parameters

1 2

2 3 1

1 1

(0)

q

q

p p p q

Y Y Y

Y Y Y

H

Y Y Y

+

+ + −

 

 

=

 

L

L

M M O M

L

 (29) 

The fundamental rule is that the Hankel matrix must be formed such that its rank is larger 

than the order of the system to be identified. In theory, the Hankel matrix H(0) and the

state-space model are related by

1 2

2 3 1

1 1

1

1

(0)

q

q

p p p q

q

p

Y Y Y

Y Y Y

H

Y Y Y

C

CA

B AB A B

CA

+

+ + −

−

−

 

 

=

 

 

 

  =      

   

L

L

M M O M

L

L M

 (30)

To determine A, B, C, first decompose the matrix H(0) by using singular value

decomposition to yield 
14

[ ] [ ]

1/ 2 1/ 2

0

(0)

0

r T T

r t r t

t

T T

r r r t t t

T

r r r

r r r r

H U V U U V V

U VUV

U V

U V

  ∑

= ∑ =     ∑

= ∑ + ∑

≈ ∑

=∑∑      

 (31)

where U and V are orthonormal matrices such that T U U I = and T V V I = , and ∑t

 is a

diagonal matrix containing the singular values that are negligible in comparison with

those in the diagonal matrix ∑r. Comparison of Eqs. (30) and (31) establishes the

following equalities

1/ 2

1/ 2

First rows of 

First columns of 

r r

r r

C m U

B r V

= ∑    

= ∑   

 (32)

The equalities in Eq. (32) are not unique, but they are balanced because both share the

matrix ∑ equally. To determine the state matrix A, another Hankel matrix must be

formed, i.e.,

2 3 1

3 4 2

1 2

(1)

q

q

p p p q

Y Y Y

Y Y Y

H

Y Y Y

+

+

+ + +

 

 

=

 

L

L

M MOM

L

 (33)

This matrix is formed by deleting the first column of H(0) and adding a new column at

the end of the matrix. As a result, H(1) has the following relationship with the system

matrices A, B, and C 

2 3 1

3 4 2

1 2

1

1

(1)

q

q

p p p q

q

p

Y Y Y

Y Y Y

H

Y Y Y

C

CA

A B AB A B

CA

+

+

+ + +

−

−

 

 

=

 

 

 

  =      

   

L

L

M MOM

L

L M

 (34)
15

Substituting Eq. (31) into Eq. (34) thus yields

† † 1/ 2 1/ 2

1/ 2 1/ 2

(1)

(1)

r r r r

T T

r r r r

A U H V

U H V − −

=  ∑ ∑       

= ∑ ∑

 (35)

The symbol † means pseudo-inverse. The orthonormal property of U and V shown in Eq. 

(31) has been used to derive Eq. (35).

Assume that the state matrix A of order n has a complete set of linearly

independent eigenvectors 1 2 , , , ψ ψ ψ L n with corresponding eigenvalues 1 2, , , λ λ λ L n

which are not necessarily distinct. Define Λ as the diagonal matrix of eigenvalues andΨ

as the matrix of eigenvectors, i.e., 

1

2

n

λ

λ

λ

 

 

Λ =  

 

   

O

 (36)

and 

Ψ = [ψ1 2 ψ ψ L n ] (37)

The identified A, B, and C can then be transformed to become Λ , 

1B− Ψ , and CΨ . The 

state-space model, Eq. (21), in modal coordinates, becomes

( 1) ( ) ( )

( ) ( ) ( )

m m m

m m

x k x k Buk

y k C x k Du k

+ = Λ +

= +

 (38)

where 

1

1

1

( ) ( ) m

m

m

A

x k x k

B B

C C

−

−

−

Λ = Ψ Ψ

= Ψ

= Ψ

= Ψ

The diagonal matrix Λ contains the modal damping rates and the damped natural

frequencies. The matrix 1 B B m

−

= Ψ defines the initial modal amplitudes and the matrix

C C m = Ψ the mode shapes at the sensor points. All the modal parameters of a dynamic 

system can thus be identified by the three matrices Λ , 

1B− Ψ , and CΨ . The desired

modal damping rates and damped natural frequencies are simply the real and imaginary 
16

parts of the eigenvalues Λc, after transformation from the discrete-time domain to the

continuous-time domain using the relation 

1

Log( ) c

t

Λ = Λ

∆

 (39)

Modal Controllability and Observability

Equation (30) shows that the Hankel matrix H(0) is formed by two matrices

defined by

1

p

p

C

CA

CA −

 

 

=  

 

   

M

O (40)

and

q 1

q B AB A B−

=   C   L (41)

The matrix Op is commonly called the observability matrix whereas Cq is referred to as

the controllability matrix. If the system possesses n states, both Op and Cq may have a

maximum rank of n assuming that the integers p and q are sufficiently large. Note that

Opand Cq may not share the same rank. If either Op or Cq is short of rank n, then some

of eigenvalues may not be identifiable. Let us elaborate on this statement by computing

Opand Cq in modal coordinates. 

The matrices Op

and Cq in modal coordinates become

1

1 1

1

m

m

p p

p p

m

C C

C C A

C C A

−

− −

−

  Ψ       Λ   Ψ    Ψ Ψ

=   =   = Ψ       Λ    

  Ψ    Ψ Ψ  

%

M M

O O (42)

and
17

1

1

1 1 1 1 1

1

q

q m m m

q

q

B B B

B A B A B

−

−

− − − − −

−

=   Λ Λ  

  = Ψ 

Ψ Ψ Ψ     Ψ Ψ Ψ    

= Ψ

% L

L

C

C

 (43)

The matrix p

%O is the modal observability matrix whereas q%

C is the modal controllability

matrix. Both matrices are coordinate dependent. Nevertheless, the Hankel matrix H(0) is

coordinate invariant,

1

(0) H p q p q p q

−

= = ΨΨ = % % O C O C O C (44)

There are n columns in Op and n rows in Cq.

Let io be the ith column of Op and T

i

c the ith row of Cq

. The Hankel matrix

H(0) can be rewritten as

1

(0)

n

T

p q i i

i

H o c

=

= = ∑

% % O C (45)

Both io and T

i

c are associated with the ith eigenvalue λi

. The matrix H(0) is the sum of n

rectangular matrices formed by the product of T

o ci i with i n = 1,K . Either i

o or T

i

c will

have contribution to H(0) only if it is not a null vector. For example, if the input actuators 

are located at the nodes of the ith mode shapes, the row vector T

i

c is null and thus its

norm is zero. A similar statement is also true for the output sensors and the corresponding 

vector io . The degree of contribution of the system eigenvalue λi to H(0) is determined 

by the product of T

o ci i which yields a pm × qr matrix. The norms oi

 and ic are

commonly used to measure the degree of observability and controllability, respectively,

for the eigenvalue λi. On the other hand, the product oi ic is a measure of the

contribution of λi

 to H(0). A relatively larger product oi ic means easier

identification of the eigenvalue λi from the input and output data. Note that the input

signal must be rich enough to excite the controllable and observable modes to be

identified.
18

Numerical Example

A system identification (SID) model for an unsteady aerodynamic flow has been

created for several wing motions or gust excitations and corresponding aerodynamic

responses. These models were derived from numerical simulations using a vortex lattice

(VL) model for a delta wing with 55 vortex elements on the wing and 300 vortex

elements in the wake (i.e. in Fig. 1, km = kn = 10, kmm = 20). In each case, the flow

about the wing is excited by a certain type of prescribed downwash at the wing points,

w(t).

The numerical VL model produced vortex strengths at the grid points of the wing

and in the wake, G (t), and the corresponding pressure distribution on the wing, p(t). These 

data were then used as input for the SID model.

The following excitations to the flow have been considered:

1) step angle of attack, w(t)=const for t>0,

2) frozen (fixed with respect to the fluid fixed coordinates) sharp edge gust, w(tx/v)=const for (t-x/v)>0, where v is the airfoil or flow velocity,

3) frozen gust of changing frequency, w(t-x/v)=const 2

max sin(ω (t − x / ) v T / 2 ), which is

sometimes called a swept gust, where the frequency of the sweep changes from zero

to the maximal frequency, wmax , within the period of the sweep, T.

4) random downwash (w at each grid point and at every time step is a random number),

5) frozen random gust (w at the first grid point [root leading edge element] is a random

number at every new time step, that w is then convected with the freestream velocity

for the following panels during next steps).

The excitation cases 1, 2, 3, and 5 are x coordinate dependent only. The excitation

signal at a specific location x is constant along the axis y as shown in Fig. 1. This implies 

that the excitation data matrices used for system identification are rank deficient, i.e., the 

rank is less than the number of inputs. The system matrices A, B, C, and D thus identified 

by using these types of excitation would not be accurate, in particular the columns in the 

input matrix B may not be independent. The number of independent columns in B is

equal to the rank of the excitation data matrix.
19

On the other hand, the random excitation defined in case 4 is both x and y coordinate 

dependent. The excitation data matrix used for identifying the system matrices A, B, C, 

and D is of full rank (i.e., the number of inputs) with the assumption that the excitation

data length is equal to or longer than the number of inputs. The input matrix B in

particular would be accurately identified in theory for a controllable and observable

system. All columns in B are linearly independent. 

Using a system identification model

The ability of a SID model obtained from one of the excitations to predict the

flow for another possible excitation was studied. The aformentioned five flow excitations 

were considered. Results were obtained for either vortex strength data or pressure data.

Consider the vortex strength data case first. Using the time histories (1000 time steps) for 

the downwash and the vortex strengths, where the latter was generated by the VL model

for the given downwash, a SID model was obtained. That is, the state-space matrices A,

B, C, and D for this particular downwash were computed. Then this SID model was used 

to predict the vortex strength histories due to another downwash and the results compared 

with the corresponding vortex strength time histories obtained by the VL model itself for 

that downwash. To quantify the differences between the VL and SID outputs an error, d, 

defined as d=100% |Q-y|/|Q| was used, where the norm |X| is defined to be the largest

singular value of X. In this case, both Q (VL vortex strengths) and y (SID predicted

vortex strengths) are 1000 x 355 rectangular matrices (1000 time steps and 355 degrees

of freedom). The same method was also used for the pressure data. There Q (VL

pressure) and y (SID predicted pressure) are 1000 x 55 (55 elements on the wing)

matrices. These results are presented in Figures 2 and 3 and in Tables 1 and 2. 

Consider Fig. 2. Here the percent errors are sorted based upon using the wing

motion or gust excitation used to obtain a SID model, while the errors correspond to how 

well that SID model predicts the vortex strength (Fig 2a) and pressure (Fig. 2b) time

histories for other excitations. A limit of 50% is used for the vertical axis scale in order to 

show small errors better. When applied to the very same downwash, the SID models

compute the output with the less than 1% error (see Ref. [2]). As seen in the figure for
20

both the vortex strengths and pressure, the random excitation case is the best choice to

obtain an SID model that will predict the behavior for any of the considered cases. In Fig 

2a one can see that, if not asked to predict the very special case of a random downwash,

three other models based on sharp edge, swept gust and frozen random excitation also

work reasonably well. 

The results in Figure 3 are sorted based upon the flow one is trying to predict with 

a SID model. For example, from the Fig. 3a one can conclude that to be able to predict

the vortex strengths due to a frozen swept or frozen random gust, SID models obtained

from any excitation case (except the step angle of attack excitation) would do a very good 

job, while for the prediction of vortex strengths due to a step angle of attack wing motion 

or frozen gust excitation any of considered SID models would perform well. Not

surprisingly (after studying Fig.2), both Fig. 3a and 3b show that only a SID model

obtained from random downwash data can predict the time history of aerodynamic flow 

due to the random downwash. The values of these errors are presented in the Tables 1

(vortex strengths) and 2 (pressure), where the rows of the tables relate to the Fig. 2 and

columns to the Fig. 3. 

It is useful to consider the number of singular values kept in the SID

models. The number following the wing motion/gust designation in the first column of

the tables is the number of singular values retained in the identified SID model for that

wing motion/gust. For the vortex strength data cases, the maximum number of singular

values varies from 50 for the sharp edge gust to 300 for the random gust. This number is 

chosen such that the ratio of the largest singular value to the smallest one kept was 12 10 . 

For the vortex strength case, the total number of singular values is 300 (which, in fact, is 

equal to the total number of panels minus the number of panels on the wing). Thus, in the 

random gust case the signal is such that all the singular values are relatively high and

close to each other magnitude.

Lift

As a global measure of the aerodynamic flow, consider the lift on the wing vs

time. See Fig. 4. In Fig. 4a, b the lift time history for the motion of the wing due to the
21

step angle of attack is presented. In Fig. 4c, d the lift time history for the excitation by the 

frozen sharp edge gust is shown. (Each case is split into two figures to show in detail the 

predicted steady state lift.) The solid line is the “original” lift (obtained from the VL

model). The left pointed and right pointed triangles represent lift time histories from SID 

models obtained from the step angle of attack motion and frozen sharp edge gust

respectively, circles – frozen swept gust, squares – random gust, and diamonds – frozen

random gust. (Note that the steady state lift due to either step angle of attack or sharp

edge gust of the same amplitude is the same. That is why Fig. 4b looks very much like

Fig. 4d.) By comparing the graphs in Fig. 4 with either results in Fig 3b or Table2, one

can conclude that the error norm previously discussed does a good job of reflecting the

difference between the “original” flow and those predicted by the identified models.

However, the additional insight gained from the plots in Fig. 4 (see Fig. 4b and d) is that 

SID models from all excitations (but the swept gust) predict the steady state flow very

well.

From the above numerical results, it was clearly shown that the system response

may not be reproduced using an identified model from excitations other than the random

gust or its own excitation. Note that only the random gust could identify an accurate

model described by the system matrices A, B, C, and D for all other inputs. Other kinds 

of excitation including the step angle, sharp edge, swept gust, and frozen random could

only reproduce its own response, although some excitations such as swept gust may

identify a model that gives a reasonable prediction for the responses produced by step

angle, sharp edge, and frozen random. Nevertheless, none of the identified models

produced by the step angle, sharp edge, swept gust, and frozen random could predict the

response excited by the random gust. This is because the excitations other than the

random gust did not excite all system modes in order to identify a state-space model to

represent the system accurately. Recall that only in the case of random gust all the

excitation (downwash) data are independent while in the four other cases only ten of

them are independent (the downwash along the y coordinate is the same). Some modes

can only be excited by independent downwash along the x coordinate as well as the y

coordinate. 
22

Modal Controllability and Observability

The modal observability and controllability matrices (42) and (43) were computed 

for small numbers of outputs, Y, and inputs, U, respectively. The ranks of the

controllability and observability matrices ranged between 63 and 91 depending on what

output and input elements were used when building (42) and (43). For example, the rank

of the controllability matrix in the case when there is only one input applied at the tip

element of the wing (only the last column of the full input influence matrix, B, remains) 

was 91. Remember the system posses n = 355 states, the maximum rank can be as much 

as 355. Therefore, because the ranks of the controllability and observability matrices are 

less than 355, the system is may not be controllable and/or observable if only one input

located at the tip element of the wing is used. Moreover, a controllability vector, bm, (see 

section 5.1.1 of Ref. [3]) was computed. This vector is formed by the product of the

inverse of the modal matrix with the eigenvectors of A as its column vectors and a

column of the input influence matrix B. It was previously shown (Ref. [3]) that if bm has 

a zero element, than the control force applied at that modal node cannot control this

mode. This means that the control force is acting exactly at the node of the mode.

It was found that the modes corresponding to the dominant branch of the

aerodynamic flow eigenvalues (for an earlier discussion of the eigenvalues see Ref. [2])

are the least controllable for the input located at the tip element of the wing. The

components of the vector bm were scaled such that the largest component is 1. In Figure 

5, the continuous time eigenvalues of the system are shown: in Fig. 5a those eigenvalues 

that have corresponding components of the controllability vector with values larger than

0.1 are noted by a cross, x; in Fig. 5b similar results are shown for 0.01. As one can see,

even in the latter case the eigenvalues of the dominant branch are still not marked. 

Values of all components of the controllability vector are shown in Figure 6a. The 

mode number is plotted along the horizontal axis, while the vertical axis shows the value 

of the components of the scaled vector bm

. The modes corresponding to the dominant 

branch of the eigenvalues are numbered from 1 to 25 (the first 10 of which correspond to 

the closest to zero circle), and as seen from Fig. 6a they are the least controllable. The
23

same result persists if other locations of the input force are considered or indeed if the

force applied at more than one location including all 55 vortex lattice panels on the wing. 

An investigation of modal observability showed that the dominant modes are the

most observable. Observability vector components are presented in Figure 6b and the

continuous time eigenvalues that correspond to modes with scaled observability vector

larger than certain value are presented in Figure 7. In Fig. 7a, the components larger than 

0.4 were crossed (note the two most right circles corresponding to first 11 modes are

crossed) and in Fig. 7b, all of the components that are larger than 0.2 were marked.

Unlike the controllablity case, the modes that correspond to the dominant branch are the

most observable.

Qualitatively the same controllability and observability results were found when

the identified system was obtained using considered excitations. Such results for the case 

of the random gust excitation are presented in Figure 8: the modes corresponding to the

dominant branch are the least controllable (Fig 8a) and the most observable (Fig 8b).

When pressure data was used to obtain A, B, and C matrices, again the system

was found to be neither controllable nor observable with only one input located at the tip 

of the delta wing, even though the scaled controllability and observability vectors look

different in this case: these vectors for the random gust excitation are presented in Figure 

9. Note, however, that the 55 modes that appear here do not correspond to those of the

VL model or at least such a relation was not established. 

Modal controllability provides a measure of how difficult it may be to excite the

respective mode by the input(s). For simplicity without losing generality, let us consider

only one input located at the tip element of the wing. A weak modal controllability

means that the considered mode is difficult to excite by the chosen input. One may

consider selecting an input location or several locations that would produce a relatively

stronger modal controllability for the modes of interest. In our example shown above, it is 

seen that the dominant branch of the eigenvalues is the least controllable in comparison

with other branches of eigenvalues regardless of the locations of the excitation inputs. 

Nevertheless, eigenvalues in the dominant branch are the most observable. In system

identification, both controllability and observability are equally important. A

combination of weak controllability and strong observability may be sufficient to identify 
24

the eigenvalues (modes) of interest. A conventional measure of the modal identifiability

is the product of the modal controllability and observability for the mode of interest. An 

experimentalist before designing and performing an experiment, may need to use an

analytical state-space model to evaluate the modal identifiability for the modes of interest 

in order to choose proper locations for excitation inputs and measurement outputs. In

addition, the excitation signal must be rich in frequency for the modes of interest to excite 

these modes properly.

Conclusions

A state-space representation of a theoretical vortex lattice model has been

developed using a system identification approach. The case of limited measurement data

has been considered. This is done in anticipation of using the proposed system

identification method with experimental data, e.g. a system identification model might be 

obtained from pressure (sensor) measurements on a wing in a wind tunnel. Two possible 

system identification approaches have been presented. Numerical results showed the

importance of the choice of the wing motion or gust under consideration on building

system identification models and predicting the vortex strengths or pressure on a wing.

Modal controllability and observability have been discussed. Numerical results for the

chosen delta wing vortex lattice model have shown that the modes corresponding to the

dominant branch of the eigenvalues of the flow are the least controllable and most

observable. By least controllable one means that these modes are difficult to excite by the 

type of downwash discussed in this paper such as the step angle of attack, frozen sharp

edge gust, frozen gust of changing frequency, random downwash, or frozen random gust. 

On the other hand, by most observable one means that the dominant modes are easy to

observe in terms of vortex strengths or the pressure distribution on the wing.

Identifiability of modes depends clearly on both modal controllability and observability.

The product of modal controllability and observability is commonly used as a measure of 

modal identifiability. 
25

References

1. Hall, K. C., “Eigenanalysis of Unsteady Flows About Airfoils, Cascades, and Wings,”

AIAA Journal, Vol. 32, No. 12, 1994, pp. 2426-2432.

2. Tang, D. M., Kholodar, D. B., Juang, J.-N., Dowell, E. H., “System Identification and 

POD Method Applied to Unsteady Aerodynamics,” accepted for publication in AIAA 

Journal, 2000.

3. Juang, J.-N., Applied System Identification, Prentice Hall, Inc., Englewood Cliffs,

New Jersey 07632, 1994, ISBN 0-13-079211-X.

Table 1: Percent errors using the vortex strength data

Step 

Angle

Sharp 

Edge

Swept 

Gust

Random 

Gust

Frozen 

Random

Step Angle, 67 0.41 3.40 246.07 84.11 79.09

Sharp Edge, 50 11.64 0.00 0.21 83.90 0.00

Swept Gust, 233 0.12 0.11 0.05 78.04 0.11

Random Gust, 300 0.00 0.00 0.00 0.00 0.00

Frozen Random, 104 13.49 0.07 0.25 89.57 0.07

Table 2: Percent errors using the pressure data

Step 

Angle

Sharp 

Edge

Swept 

Gust

Random 

Gust

Frozen 

Random

Step Angle, 17 0.05 4.25 168 83.8 90.7

Sharp Edge, 35 20.9 0.66 0.37 320 0.65

Swept Gust, 52 15.3 15.4 2.69 72.9 15.3

Random Gust, 110 0.42 0.42 0.07 0.39 0.42

Frozen Random,45 25.9 1.48 0.74 290 1.46
26

air flow wing

elements

wake 

elements

horseshoe

vortex

X

Y

kn

km kmm

Figure 1: Vortex lattice model of unsteady 
ow about a delta wing.
27

0

25

50

(a)

1

1

1 1

1

2

2 2 2 2

3

3 3 3 3

4 4 4

4

5 4

5 5 5 5

step angle sharp edge swept gust random frozen random

0

25

50

(b)

1

1

1

1

1

2

2

2

2 2

3

3

3

3 3

4 4 4

4

5 4

5

5

5 5

step angle sharp edge swept gust random frozen random

Figure 2: Percent error for (a) vortex strength and (b) pressure time histories for the ve

described excitations (marked by the numbers). The SID models used to simulate these

histories are indicated along the horizontal axis.
28

0

25

50

(a)

1

1

1 1

1

2

2 2

2

3 3 3 2

3

4 4 4 4 34

5

5 5

5

5

step angle sharp edge swept gust random frozen random

0

25

50

(b)

1

1

1 1 1

2

2 2

2

2

3 3

3

3

3

4 4 4 4 4

5

5 5

5

5

step angle sharp edge swept gust random frozen random

Figure 3: Percent error for (a) vortex strength and (b) pressure time histories that were

simulated by the SID models (marked by the numbers). The 
ows under consideration

are indicated along the horizontal axis.
29

0 2 4 6 8 10

−1

−0.5

0

0.5

1

1.5

2

2.5

3

(a)

Vortex Lattice

step angle 

sharp edge 

swept gust 

random gust 

frozen random 

190 192 194 196 198 200

1

1.05

1.1

1.15

1.2

1.25

1.3

(b)

Vortex Lattice

step angle 

sharp edge 

swept gust 

random gust 

frozen random 

0 2 4 6 8 10

0

0.5

1

1.5

2

(c)

Vortex Lattice

step angle 

sharp edge 

swept gust 

random gust 

frozen random 

190 192 194 196 198 200

1

1.05

1.1

1.15

1.2

1.25

1.3

(d)

Vortex Lattice

step angle 

sharp edge 

swept gust 

random gust 

frozen random 

Figure 4: Lift time history for the excitations of (a and b) step angle of attack and (c and

d) frozen sharp edge gust.
30

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0

0

2

4

6

8

10

12

14

16

Real Part of Eigenvalues

Imaginary Part of Eigenvalues

(a)

−−− dominant

 branch 

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0

0

2

4

6

8

10

12

14

16

Real Part of Eigenvalues

Imaginary Part of Eigenvalues

(b)

−−− dominant

 branch 

Figure 5: System eigenvalues. The crossed eigenvalues correspond to modes for which

components in the scaled controllability vector are larger then (a) 0.1, (b) 0.01. The

largest controllability vector component is 1 (cf. Fig. 6a).
31

0 50 100 150 200 250 300 350

0

0.2

0.4

0.6

0.8

1

Mode Number

Components of the Controlability Vector

(a)

0 50 100 150 200 250 300 350

0

0.2

0.4

0.6

0.8

1

Mode Number

Components of the Observability Vector

(b)

Figure 6: Contollability (a) and observability (b) vector components. In this case, the

state, A, and input in
uence, B, matrices were derived in the close form from the VL

model with the use of Eqs. (7), where the output in
uence matrix, C, is an identity

matrix. The only input is located at the tip of the delta wing.
32

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0

0

2

4

6

8

10

12

14

16

Real Part of Eigenvalues

Imaginary Part of Eigenvalues

(a)

−−− dominant

 branch 

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0

0

2

4

6

8

10

12

14

16

Real Part of Eigenvalues

Imaginary Part of Eigenvalues

(b)

−−− dominant

 branch 

Figure 7: System eigenvalues. The crossed eigenvalues correspond to modes for which

components in the scaled observability vector are larger then (a) 0.4, (b) 0.2. The largest

observability vector component is 1 (cf. Fig. 6b).
33

0 50 100 150 200 250 300

0

0.2

0.4

0.6

0.8

1

Mode Number

Components of the Controlability Vector

(a)

0 50 100 150 200 250 300

0

0.2

0.4

0.6

0.8

1

Mode Number

Components of the Observability Vector

(b)

Figure 8: Contollability (a) and observability (b) vector components. In this case, matrices A, B, and C were identied from the vortex strength data when the wing was excited

by the random gust. The only input is located at the tip of the delta wing.
34

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

Mode Number

Components of the Controlability Vector

(a)

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

Mode Number

Components of the Observability Vector

(b)

Figure 9: Contollability (a) and observability (b) vector components. In this case, matrices A, B, and C were identied from the pressure strength data when the wing was

excited by the random gust. The only input is located at the tip of the delta wing.
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE

October 2001

3. REPORT TYPE AND DATES COVERED

Technical Memorandum

4. TITLE AND SUBTITLE

System Identification of a Vortex Lattice Aerodynamic Model

5. FUNDING NUMBERS

WU 706-32-21-01

6. AUTHOR(S)

Jer-Nan Juang, Denis Kholodar, and Earl H. Dowell 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

8. PERFORMING ORGANIZATION

REPORT NUMBER

 L-17897

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA/TM-2001-211229

11. SUPPLEMENTARY NOTES

 

12a. DISTRIBUTION/AVAILABILITY STATEMENT

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12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The state-space presentation of an aerodynamic vortex model is considered from a classical and system

identification perspective. Using an aerodynamic vortex model as a numerical simulator of a wing tunnel

experiment, both full state and limited state data or measurements are considered. Two possible approaches for

system identification are presented and modal controllability and observability are also considered. The theory

then is applied to the system identification of a flow over an aerodynamic delta wing and typical results are

presented.

14. SUBJECT TERMS

System Identification; Vortex Lattice Model; Proper Orthogonal Decomposition;

15. NUMBER OF PAGES

39

Reduced Aerodynamic Model 16. PRICE CODE

A03

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