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VECTOR DIFFERENTIAL EQUATIONS 281
6.5.2 Discretization of LTI State Equation
In this section, we consider a discretization method of converting a continuous-
time LTI (linear time-invariant) state equation

x (t) = Ax(t) + Bu(t) with the initial state x(0) and the input u(t) (6.5.13)
into an equivalent discrete-time LTI state equation with the sampling period T

x[n + 1] = A d x[n] + B d u[n] (6.5.14)
with the initial state x[0] and the input u[n] = u(nT ) for nT ≤ t< (n + 1)T
which can be solved easily by an iterative scheme mobilizing just simple multi-
plications and additions.
For this purpose, we rewrite the solution (6.5.8) of the continuous-time LTI
state equation with the initial time t 0 as
t

x(t) = φ(t − t 0 )x(t 0 ) + φ(t − τ)Bu(τ) dτ (6.5.15)
t 0
Under the assumption that the input is constant as the initial value within each
sampling interval—that is, u[n] = u(nT ) for nT ≤ t< (n + 1)T —we substitute
t 0 = nT and t = (n + 1)T into this equation to write the discrete-time LTI state
equation as
(n+1)T
x((n + 1)T ) = φ(T )x(nT ) + φ((n + 1)T − τ)Bu(nT ) dτ
nT
(n+1)T
x[n + 1] = φ(T )x[n] + φ(nT + T − τ) dτBu[n]
nT
x[n + 1] = A d x[n] + B d u[n] (6.5.16)

where the discretized system matrices are
A d = φ(T ) = e AT (6.5.17a)

(n+1)T 0 T
σ=nT +T −τ
B d = φ(nT + T − τ) dτB = − φ(σ) dσB = φ(τ) dτB
nT T 0
(6.5.17b)
Here, let us consider another way of computing these system matrices, which
is to the taste of digital computers. It comes from making use of the definition
of a matrix exponential function in Eq. (6.5.6) to rewrite Eq. (6.5.17) as

∞ m m ∞ m m
A T A T
AT
A d = e = = I + AT = I + AT (6.5.18a)
m! (m + 1)!
m=0 m=0
T ∞ m m ∞ m m+1
A τ A T
T
B d = φ(τ) dτB = dτB = B = TB (6.5.18b)
0 0 m! (m + 1)!
m=0 m=0