Sampled data systems. Discrete

Sampled data systems.
Discrete‐time equivalents.
Martin Hromcik.
Automatic Control 2012.
12‐IV‐12
Simple discretization procedures.
Automatické řízení - Kybernetika a robotika
digital
Cont.time
controller
ctrl law
shaper
forward Euler rule (explicit)
dx(t ) x(t + h) − x(t )
z −1
≈
sx ≈
x
dt
h
h
sh
z = e ≈ 1 + sh
z −1
s≈
h
backward Euler rule (implicit)
dx(t ) x(t ) − x(t − h)
≈
dt
h
z = e sh ≈
Michael Šebek
z −1
sx ≈
x
zh
1
1 − sh
ARI‐21‐2011
z −1
s≈
zh
2
Tustin method
Automatické řízení - Kybernetika a robotika
bilinear transformation
2 z −1
s≈
h z +1
1 + sh 2
z=e ≈
1 − sh 2
sh
(... compare to 1st order Pade
approximation of time delay ...)
>> c2d(f,h,'tustin')
replace s with respective prescription
‐ suitable for simple pen‐and‐paper situations
‐ s‐plane mappings:
forward rule
backward rule
Tustin
3
‐ mind state space implications (A quite clear, B is affected as well though!)
Michael Šebek
ARI‐21‐2011
Discrete-time PID controllers (PSD)
Automatické řízení - Kybernetika a robotika
P:
u(t ) = Ke(t )
u(s) = Ke(s)
u( z) = Ke( z)
u(k ) = Ke(k )
I: forward rule
K t
u(t ) = ∫ e(τ )dτ
TI 0
K
u ( s) =
e(s)
TI s
K h
u( z ) =
e( z)
TI z −1
Kh
u(k + 1) = u(k ) +
e(k )
TI
D: backward rule
u(t ) = KTDe&(t )
u(s) = KTD se(s)
KTD
z −1
u(k + 1) =
u( z) = KTD
e( z)
( e(k +1) − e(k ))
h
zh
Michael Šebek
ARI‐21‐2011
4
Discrete-time PID controllers (PSD)
Automatické řízení - Kybernetika a robotika
• PSD
⎡ h 1 TD z −1⎤
+
u( z) = K ⎢1 +
⎥ e( z)
⎣ TI z −1 h z ⎦
• Alternatives:
⎡
⎤
1
u(s) = K ⎢1 +
+ TD s ⎥ e(s)
⎣ TI s
⎦
Michael Šebek
ule
r
.
fw d
Tu
stin
⎡ h z TD z −1⎤
+
u( z) = K ⎢1 +
⎥ e( z)
⎣ TI z −1 h z ⎦
⎡
h z + 1 2TD z −1⎤
+
u( z) = K ⎢1 +
⎥ e( z)
⎣ 2TI z −1 h z + 1⎦
ARI‐21‐2011
5
State feedback & discretization
Automatické řízení - Kybernetika a robotika
x& = Ax + Bu , y = Cx
u (t ) = MuC (t ) − Kx(t )
K dis = K ( I + ( A − BK ) h 2 )
M dis = ( I − KB h 2 ) M
-better correspondance of CL dynamics (in CT and DT) ...
-cont.time:
A, B
A-BK
-discretization, Tustin rule:
(I-Ah/2) \ (I+Ah/2), (I-Ah/2) \ B
(I-Ah/2) \ (I+Ah/2) - (I-Ah/2) \ B K(I+(A-BK)h/2) =
= (I-Ah/2) \ (I+Ah/2 – BK(I+(A-BK)h/2)
= (I-Ah/2) \ (I+Ah/2 – BK – BKAh/2 – BKBKh/2) ...
Michael Šebek
ARI‐21‐2011
6
ZOH discretization.
Automatické řízení - Kybernetika a robotika
r( z)
D( z)
r (t )
r (kh)
h
digital
Diferenční
u(kh)
rovnice
ctrl law
D/A
D/A
a
tvarovač
shaper
u(t )
G (s)
y(t )
e(kh)
clock
hodiny
senzor
A/D
Michael Šebek
y(t )
y(kh)
h
ARI‐21‐2011
1
G( z)
7
Systémy
ZOH
a řízení
u(s)
ZOH
u(kh)
G(s)
u(t)
y(s)
Y(s)
y(t)
u( z)
u(kh)
y(kh)
G( z )
y( z)
y(kh)
u(kh) = 1 k = 0
u(kh) = 0 k ≠ 0
ZOH
ZOH
ZOH impulse
impulse response
response
1 1 − hs
− e
s s
u(t ) = 1(t ) −1(t − h)
system dynamics
G(s)
Y (s) = (1− e )
s
−hs
Michael Šebek - ČVUT - 2006
8
Systémy
Discrete-time description
a řízení
u(s)
notation
G ( z ) = Z {_ y (kT )}
ZOH
= Z {_L−1 {Y ( s )}} = Z {_Y ( s )}
{
= Z _(1 − e − hs )
G(s)
s
}
{ } {
G(s)
G ( s)
G( z) = Z _
− Z _e−Ts
s
s
{
Z _e − hs
}
u(kh)
G(s)
u(t)
u( z)
}
u(kh)
y(s)
Y(s)
y(t)
G( z )
y(kh)
y( z)
y(kh)
{ }
G(s)
G(s)
== z −1Z _
s
s
{ }
G ( s)
G ( z ) = (1 − z −1 ) Z _
s
Michael Šebek - ČVUT - 2006
c2d(G,h,'zoh')
c2d(G,h)
9
Systémy
ZOH & state-space
u(t)
u(kh)
ZOH
x& = Ax + Bu
y = Cx + Du
y(t)
Y(s)
a řízení
u(kh)
y(kh)
xk +1 = Φxk + Γuk
yk = Cxk + Duk
y(kh)
x& = Ax + Bu
y = Cx + Du
x(t ) = e
A ( t − t0 )
t
x(t0 ) + ∫ e A(t −τ ) Bu (τ )dτ
Michael Šebek - ČVUT - 2006
t0
10
Systémy
ZOH & state-space
tk
a řízení
tk +1
uk = u (τ ),τ ∈ [tk , tk +1 )
h = tk +1 − tk
x(tk +1 ) = e
A ( tk +1 −tk )
x(tk ) + ∫
tk +1
tk
= e x(tk ) + ∫
Ah
tk +1
tk
= e x(tk ) +
Ah
(∫
h
0
e A(tk +1 −τ ) Bu (τ )dτ
e A(tk +1 −τ ) dτ Bu (tk )
)
ν = tk +1 − τ
e Aν dν Bu (tk )
x(tk +1 ) = Φx(tk ) + Γu (tk )
y (tk ) = Cx(tk ) + Du (tk )
Michael Šebek - ČVUT - 2006
Φ = eAh
Γ=
(∫
h
0
)
eAν dν B
11