Digital Control Systems [MEE 4003] - Kckong.info

2.1. STATE SPACE REALIZATION OF DYNAMIC SYSTEMS 21
Solution of diagonalizable state space models
For a diagonalizable matrix F = VΛV −1 ∈ R n×n , where Λ is a diagonal matrix with the
eigenvalues ofF ,e Ft = e VΛV −1t is calculated by the Taylor expansion as follows.
e (VΛV −1 )t
= I +VΛV −1 t+ 1 2 VΛV −1 VΛV −1 t 2 + 1 6 VΛV −1 VΛV −1 VΛV −1 t 3 +...
[
= V I +Λt+ 1 2 Λ2 t 2 + 1 ]
6 Λ3 t 3 +... V −1
[ ∞
]
∑ 1
= V
i! Λi t i V −1
i=0
= Ve Λt V −1
Note thatVV −1 has been canceled in the equation above. Note thate Λt is
⎡ ⎤
e λ 1t
0 0
e Λt 0 e λ 2t
0
= ⎢
⎣
. ..
⎥
⎦
0 0 e λnt
Therefore, the exponential of diagonalizable matrices can be solved as
⎡ ⎤
e λ 1t
0 0
e Ft 0 e λ 2t
0
= V ⎢
⎣
. ..
⎥
⎦ V −1 (2.5)
0 0 e λnt
From the result in (2.5), the solution to diagonalizable state space models can be
obtained as follows. Suppose a state space model is given:
ẋ = Fx+Gu
y = Hx
x(0) = x 0
where F is a diagonalizable matrix in R n×n such that F can be eigendecomposed to
F = VΛV −1 , whereV is a matrix consisting of the eigenvectors ofF . Then, a new state
is defined as
¯x = V −1 x ∈ R n
orx = V ¯x. SubstitutingV ¯x forx, the state space model becomes
V ˙¯x = FV ¯x+Gu
y = HV ¯x
¯x(0) = V −1 x 0
Digital Control Systems, Sogang University
Kyoungchul Kong