Page 206 - Numerical Analysis Using MATLAB and Excel

Summary

where for two or more simultaneous differential equations and are 2 × 2 or higher order
A
C
matrices, and and are column vectors with two or more rows, the solution of the matrix
d
b
·
differential equation x = Ax + bu with initial conditions xt () = x 0 is obtained from the
0
relation
(
At – t ) 0 At t – Aτ
xt() = e x + e ∫ e bu τ() τ
d
0
t
0
At
where the state transition matrix e is defined as the matrix power series
1
1
1
n n
2 2
3 3
ϕ t() ≡ e At = I + At + -----A t + ----A t + … + -----A t
-
2! 3! n!
and is the n × n identity matrix.
I
,,
λ
• If is an n × n matrix, and be the n × I n identity matrix, the eigenvalues , i = 1 2 … n of
,
A
i
A are the roots of the nth order polynomial
]
[
det A λI = 0
–
• Evaluation of the state transition matrix e At is based on the Cayley−Hamilton theorem. This
(
)
theorem states that a matrix can be expressed as an n – 1 th degree polynomial in terms of the
matrix as
A
2
e At = a I + a A + a A + … + a n – 1 A n – 1
2
0
1
where the coefficients are functions of the eigenvalues .
λ
a
i
• If λ ≠ 1 λ ≠ 2 λ ≠ 3 … λ ≠ n , that is, if all eigenvalues of a given matrix are distinct, the coeffi-
A
cients are found from the simultaneous solution of the following system of equations:
a
i
2
1
a + a λ + a λ + … + a n – 1 λ n – 1 = e λ t
1
1
2
0
1
1
2
2
a + a λ + a λ + … + a n – 1 λ n – 1 = e λ t
2
2
1
2
2
0
…
2
n
λ
a + a λ + a λ + … + a n – 1 n n – 1 = e λ t
0
2 n
1 n
•If the polynomial of det A λI–[ ] = 0 has roots, and m of these roots are equal, that is, if
n
a
λ = λ = λ … 3 λ = m , λ m + 1 , λ n , the coefficients of the state transition matrix
1
i
2
At 2 n – 1
e = a I + a A + a A + … + a n – 1 A
1
0
2
Numerical Analysis Using MATLAB® and Excel®, Third Edition 5−45
Copyright © Orchard Publications