Solutions Manual For Aircraft Dynamics From Modeling to Simulation 1st Edition By Napolitano

Problems for Chapter VIII


Problem 8.1
Consider a linear dynamic system described by the following Differential Equation (DE)
with constant coefficients:
y (t )  5 y(t )  7 y(t )  3 y(t )  u(t )
Find the state variable model associated with this system. Verify that the eigenvalues of
the state matrix A are coincident with the roots of the characteristic equation of the above
system.


Solution of Problem 8.1.
The system is a 3rd order system with 1 input and 1 output  n  3, m  1, l  1 . A suitable

selection of state variables is given by:
x1 (t )  y (t )
d y (t )
x2 (t )   y (t )
dt
d 2 y (t )
x3 (t )   y (t )  x3 (t )  y (t )
dt 2
Using the system DE equation,
x3 (t )  y (t )  5 y(t )  7 y(t )  3 y(t )  u(t )
Rewriting the above equations using the state variables will lead to the following:
x1 (t )  y (t )
x1 (t )  x2 (t )
x2 (t )  x3 (t )
x3 (t )  5 x3 (t )  7 x2 (t )  3 x1 (t )  u (t )


Thus, the state variable model in a matrix format is given by:
 x1 (t )   0 1 0   x1 (t )  0
    
 x2 (t )    0 0 1   x2 (t )   0  u (t )
 x (t )   3 7 5  x (t )  1 
 3    3   

,The characteristic equation is given by:

s3  5s 2  7s  3  0
with roots :
1  1, 1  1, 1  3
The eigenvalues of the system matrix are,


 0 1 0 
 
eig   0 0 1    1, 1, 3
  3 7 5 
 
The SV output equation is given by,


 x1 (t ) 
 
y  1 0 0  x2 (t )   0u (t )
 x (t ) 
 3 

,Problem 8.2
Consider a linear dynamic system described by the following set of Differential
Equations (DEs) with constant coefficients:

 y1 (t )  5 y2 (t )  u1 (t )

 y2 (t )  3 y1 (t )  2 y2 (t )  u2 (t )
- Find the state variable model of the above system.

- The system is subjected to 2 inputs  u1 (t ), u2 (t )  and has two measurable outputs

 y1 (t ), y2 (t )  . Therefore, a transfer function-based modeling of the system will require
the evaluation of a (2 x 2) matrix of transfer functions. Find the matrix of transfer
functions under the following conditions:
- Derivation directly from the DEs
- Derivation from the state variable model.


Solution of Problem 8.2
State Variable Model
The system is a 2nd order system with 2 inputs and 2 outputs  n  2, m  2, l  2  . A

suitable selection of state variables is given by:
x1 (t )  y1 (t )  x1 (t )  y1 (t )
x2 (t )  y2 (t )  x2 (t )  y2 (t )
Using the system DE equation,

 x1 (t )  5 x2 (t )  u1 (t )

 x2 (t )  3x1 (t )  2 x2 (t )  u2 (t )
Rearranging:


 x1 (t )   5 x2 (t )  u1 (t )

 x2 (t )  3x1 (t )  2 x2 (t )  u2 (t )
Thus, the SV state equation is given by:

,  x1 (t )  0 5  x1 (t )  1 0  u1 (t ) 
      
 x2 (t )  3 2  x2 (t )  0 1  u2 (t ) 
The SV output equation is given by,
 y1 (t )  1 0  x1 (t )  0 0  u1 (t ) 
      
 y2 (t )  0 1   x2 (t )  0 0 u2 (t ) 


Derivation of Transfer Function (TF) matrix DIRECTLY from the DEs
Following the application of the Laplace transformation to the above system we have:

 sY1 ( s)  5Y2 ( s)  U1 ( s)

 sY2 ( s)  3Y1 ( s)  2Y2 ( s)  U 2 ( s)
leading to the following matrix format:

s 5  Y1 ( s)  U1 ( s) 
 3 ( s  2)  Y ( s)   U ( s) 
  2   2 
By invoking the „superimposition of effect‟ property of linear systems, we can solve the

above system first considering u2 (t )  0  U 2 (s)  0 and, next, considering

u1 (t )  0  U1 ( s)  0 . The goal is to find all the Gij ( s) transfer functions in the
relationship:
Y1 ( s)  G11 ( s) U1 ( s)  G12 ( s) U 2 ( s)
Y2 ( s)  G21 ( s) U1 ( s)  G22 ( s) U 2 ( s)
Therefore, consider the system:

s 5  Y1 ( s)  U1 ( s) 
 3 ( s  2)  Y ( s)    0 
  2   


Using Cramer‟s rule we have: