Embedded Systems
Martina Maggio – Lecture 04: continuous- to discrete-time
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Continuous-Time Linear Time-Invariant System
Process
u (t)
y (t)
ẋ (t) = A x (t) + B u (t)
𝒫∶{
y (t) = C x (t) + D u (t)
Continuous-Time Linear Time-Invariant System
Process
u (t)
y (t)
ẋ (t) = A x (t) + B u (t)
𝒫∶{
y (t) = C x (t) + D u (t)
Continuous-Time Linear Time-Invariant System
Process
u (t)
y (t)
ẋ (t) = A x (t) + B u (t)
𝒫∶{
y (t) = C x (t) + D u (t)
Continuous-Time Linear Time-Invariant System
Process
u (t)
y (t)
ẋ (t) = A x (t) + B u (t)
𝒫∶{
y (t) = C x (t) + D u (t)
We measure
the output
every 𝜋 time
units (sample)
Continuous-Time Linear Time-Invariant System
Process
u (t)
y (t)
ẋ (t) = A x (t) + B u (t)
𝒫∶{
y (t) = C x (t) + D u (t)
We apply a
new input every
𝜋 time units
(hold)
We measure
the output
every 𝜋 time
units (sample)
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Sampling
s (t)
t
Sampling
s (t)
Sampling Period 𝜋
t
Sampling
s (t)
k=0
Sampling Period 𝜋
k=1
k=2
k=3
t
Sampling
s (t)
k=0
Sampling Period 𝜋
k=1
k=2
k=3
s (0) = s [0]
t
Sampling
s (t)
k=0
Sampling Period 𝜋
k=1
k=2
k=3
s (𝜋) = s [1]
s (0) = s [0]
t
Sampling
s (t)
k=0
Sampling Period 𝜋
k=1
k=2
k=3
s (𝜋) = s [1]
s (0) = s [0]
s (2𝜋) = s [2]
t
Sampling
s (t)
k=0
Sampling Period 𝜋
k=1
k=2
k=3
s (𝜋) = s [1]
s (0) = s [0]
s (2𝜋) = s [2]
s (3𝜋) = s [3]
t
Sampling
s (t)
k=0
Sampling Period 𝜋
k=1
k=2
k=3
s (k 𝜋) = s [k]
s (𝜋) = s [1]
s (0) = s [0]
s (2𝜋) = s [2]
s (3𝜋) = s [3]
t
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Sampling and Modeling
From continuous- to discrete-time (linear time-invariant) models
We would like to take our continous-time model and transform it into a
discrete-time equivalent model, that describes the behavior of the system at the
sampling instants {k, k − 1, … , 0}
ẋ (t) = A x (t) + B u (t)
→
?
⏟⏟⏟
y (t) = C x (t) + D u (t)
Discrete-Time Model
⏟⎴⎴⎴⎴⎴⎴⎴⎴⏟⎴⎴⎴⎴⎴⎴⎴⎴⏟
Continuous-Time Model
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Sampling and Modeling
From continuous- to discrete-time (linear time-invariant) models
ẋ (t) = A x (t) + B u (t)
y (t) = C x (t) + D u (t)
The solution1 for this system is
t
x (t) = e
A(t−0)
x (0) + ∫ e A(t−𝜏) B u (𝜏) d𝜏
0
and this gives
t
y (t) = C e
A(t−0)
x (0) + ∫ C e A(t−𝜏) B u (𝜏) d𝜏 + D u (t)
0
1 For
the full derivation, see http://control.ucsd.edu/mauricio/courses/mae280a/lecture5.pdf
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Sampling and Modeling
Assumption: in between
sampling
[
) instants, we keep the value of u constant (we
hold u), i.e., ∀t ∈ k 𝜋, (k + 1) 𝜋 , u (t) = u [k]
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Sampling and Modeling
Assumption: in between
sampling
[
) instants, we keep the value of u constant (we
hold u), i.e., ∀t ∈ k 𝜋, (k + 1) 𝜋 , u (t) = u [k]
1. Solve the equation from time k 𝜋 to time t (with constant u)
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Sampling and Modeling
Assumption: in between
sampling
[
) instants, we keep the value of u constant (we
hold u), i.e., ∀t ∈ k 𝜋, (k + 1) 𝜋 , u (t) = u [k]
1. Solve the equation from time k 𝜋 to time t (with constant u)
t
x (t) = e A (t−k 𝜋) x (k 𝜋) + ∫
e A(t−𝜏) B u (𝜏) d𝜏
k𝜋
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Sampling and Modeling
Assumption: in between
sampling
[
) instants, we keep the value of u constant (we
hold u), i.e., ∀t ∈ k 𝜋, (k + 1) 𝜋 , u (t) = u [k]
1. Solve the equation from time k 𝜋 to time t (with constant u)
t
x (t) = e A (t−k 𝜋) x (k 𝜋) + ∫
e A(t−𝜏) B u (𝜏) d𝜏
k𝜋
t
x (t) = e A (t−k 𝜋) x (k 𝜋) + ∫
e A(t−𝜏) d𝜏 B u (k 𝜋)
k𝜋
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Sampling and Modeling
Assumption: in between
sampling
[
) instants, we keep the value of u constant (we
hold u), i.e., ∀t ∈ k 𝜋, (k + 1) 𝜋 , u (t) = u [k]
1. Solve the equation from time k 𝜋 to time t (with constant u)
t
x (t) = e A (t−k 𝜋) x (k 𝜋) + ∫
e A(t−𝜏) B u (𝜏) d𝜏
k𝜋
t
x (t) = e A (t−k 𝜋) x (k 𝜋) + ∫
e A(t−𝜏) d𝜏 B u (k 𝜋)
k𝜋
0
x (t) = e A (t−k 𝜋) x (k 𝜋) + ∫
−e Av dv B u (k 𝜋)
t−k 𝜋
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Sampling and Modeling
Assumption: in between
sampling
[
) instants, we keep the value of u constant (we
hold u), i.e., ∀t ∈ k 𝜋, (k + 1) 𝜋 , u (t) = u [k]
1. Solve the equation from time k 𝜋 to time t (with constant u)
t
x (t) = e A (t−k 𝜋) x (k 𝜋) + ∫
e A(t−𝜏) B u (𝜏) d𝜏
k𝜋
t
x (t) = e A (t−k 𝜋) x (k 𝜋) + ∫
e A(t−𝜏) d𝜏 B u (k 𝜋)
k𝜋
0
x (t) = e A (t−k 𝜋) x (k 𝜋) + ∫
−e Av dv B u (k 𝜋)
t−k 𝜋
t−k 𝜋
x (t) = e
A (t−k 𝜋)
e Av dv B u (k 𝜋)
x (k 𝜋) + ∫
0
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Sampling and Modeling
Assumption: in between
sampling
[
) instants, we keep the value of u constant (we
hold u), i.e., ∀t ∈ k 𝜋, (k + 1) 𝜋 , u (t) = u [k]
1. Solve the equation from time k 𝜋 to time t (with constant u)
t−k 𝜋
A (t−k 𝜋)
x (t) = e
x (k 𝜋) + ∫
e Av dv B u (k 𝜋)
⏟⏟⏟
0
Φ(t,k 𝜋)
⏟⎴⎴⎴⏟⎴⎴⎴⏟
Γ(t,k 𝜋)
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Sampling and Modeling
Assumption: in between
sampling
[
) instants, we keep the value of u constant (we
hold u), i.e., ∀t ∈ k 𝜋, (k + 1) 𝜋 , u (t) = u [k]
1. Solve the equation from time k 𝜋 to time t (with constant u)
t−k 𝜋
A (t−k 𝜋)
x (t) = e
x (k 𝜋) + ∫
e Av dv B u (k 𝜋)
⏟⏟⏟
0
Φ(t,k 𝜋)
⏟⎴⎴⎴⏟⎴⎴⎴⏟
Γ(t,k 𝜋)
2. Impose periodic sampling, t = (k + 1) 𝜋
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Sampling and Modeling
Assumption: in between
sampling
[
) instants, we keep the value of u constant (we
hold u), i.e., ∀t ∈ k 𝜋, (k + 1) 𝜋 , u (t) = u [k]
1. Solve the equation from time k 𝜋 to time t (with constant u)
t−k 𝜋
A (t−k 𝜋)
x (t) = e
x (k 𝜋) + ∫
e Av dv B u (k 𝜋)
⏟⏟⏟
0
Φ(t,k 𝜋)
⏟⎴⎴⎴⏟⎴⎴⎴⏟
Γ(t,k 𝜋)
2. Impose periodic sampling, t = (k + 1) 𝜋
k 𝜋+𝜋−k 𝜋
(
)
A (k 𝜋+𝜋−k 𝜋)
x (k + 1) 𝜋 = e
x (k 𝜋) + ∫
e Av dv B u (k 𝜋)
⏟⎴⎴⏟⎴⎴⏟
(
)
0
Φ (k+1) 𝜋,k 𝜋
⏟ ⎴⎴⎴⎴
⏟ ⎴⎴⎴⎴
⏟
(
)
Γ (k+1) 𝜋,k 𝜋
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Sampling and Modeling
From continuous- to discrete-time (linear time-invariant) models
k 𝜋+𝜋−k 𝜋
(
)
A (k 𝜋+𝜋−k 𝜋)
x (k + 1) 𝜋 = e
x (k 𝜋) + ∫
e Av dv B u (k 𝜋)
⏟⎴⎴⏟⎴⎴⏟
(
)
0
Φ (k+1) 𝜋,k 𝜋
⏟ ⎴⎴⎴⎴
⏟ ⎴⎴⎴⎴
⏟
(
)
Γ (k+1) 𝜋,k 𝜋
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Sampling and Modeling
From continuous- to discrete-time (linear time-invariant) models
k 𝜋+𝜋−k 𝜋
(
)
A (k 𝜋+𝜋−k 𝜋)
x (k + 1) 𝜋 = e
x (k 𝜋) + ∫
e Av dv B u (k 𝜋)
⏟⎴⎴⏟⎴⎴⏟
(
)
0
Φ (k+1) 𝜋,k 𝜋
⏟ ⎴⎴⎴⎴
⏟ ⎴⎴⎴⎴
⏟
(
)
Γ (k+1) 𝜋,k 𝜋
We can rewrite this as
𝜋
(
)
A𝜋
x (k + 1) 𝜋 = e
x (k 𝜋) + ∫ e Av dv B u (k 𝜋)
⏟⏟⏟
0
Φ(𝜋)
⏟⎴⎴⏟⎴⎴⏟
Γ(𝜋)
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Sampling and Modeling
From continuous- to discrete-time (linear time-invariant) models
𝜋
(
)
A𝜋
x (k + 1) 𝜋 = e
x (k 𝜋) + ∫ e Av dv B u (k 𝜋)
⏟⏟⏟
0
Φ(𝜋)
⏟⎴⎴⏟⎴⎴⏟
Γ(𝜋)
Using the discrete-time notation, we can write
x [k + 1] = Φ (𝜋) x [k] + Γ (𝜋) u [k]
𝜋
where Φ (𝜋) = e A 𝜋 and Γ (𝜋) = ∫0 e Av dv B
And calculate the output as
y [k] = C x [k] + D u [k]
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Sampling and Modeling
From continuous- to discrete-time (linear time-invariant) models
It is possible to compute Φ and Γ simultaneously, using the formula
[
Φ Γ
] = eS𝜋
0 I
where S = [
A B
]
0 0
The matrix exponential can be computed using the exp Julia or expm Matlab
function
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Sampling Period Choice
The sampling period should be related to the physics of your system:
⨳ If your system state vector changes very fast you will need very frequent
samples to properly reconstruct the changes
⨳ If the changes are very slow you can afford sampling less frequently
As a rule of thumb, you would like to have between 4 and 20 samples during the
rise time of your step response, or alternatively 𝜋 ∈ (0.1 ∕𝜔n , 0.6 ∕𝜔n ) where 𝜔n
is the natural frequency of your system (max absolute value of the eigenvalues of
A)
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