Control Systems 2
Lecture 13: Digital controller design
Roy Smith
2016-5-24
13.1
Digital control system design
Sampled-data closed-loop
y(k)
y(t)
T
G(s)
u(t)
ZOH
u(k)
Kd (z)
+
−
Kd (z)
+
r(k)
GZOH (z) equivalence
GZOH (z)
y(k)
T
G(s)
ZOH
r(k)
−
2016-5-24
13.2
Zero-order hold equivalence — transfer function
y(k)
y(t)
G(s)
T
|
Input:
u(k) =
Output:
(
1
0
u(t)
{z
}
GZOH (z)
k = 0,
,
k 6= 0
u(k)
ZOH
u(t) = step(t) − step(t − T ).
G(s)
.
y(s) = 1 − e−T s
s
We now sample this, and take the Z-transform,
−T s G(s)
GZOH (z) = Z
1−e
s
G(s)
−1
= (1 − z )Z
.
s
2016-5-24
13.3
Zero-order hold equivalence — state space
y(k)
y(t)
G(s)
T
|
u(t)
u(k)
ZOH
{z
}
GZOH (z)
Integrating Φ(t) over a single sample period (kT to kT + T ):
Z kT +T
AT
x(kT + T ) = e x(kT ) +
eA(kT +T −τ ) Bu(τ ) dτ,
kT
2016-5-24
A
C
B
D
ZOH-equivalence
=⇒

AT
 e

C
Z
T
Aη
e
0
D

B dη 

13.4
Zero-order hold equivalence — frequency domain
Example:
G(s) =
(2 − s)
,
(2s + 1)(s + 2)
T = 0.6
0.1
π/T
1
10
1
log ω
(rad/sec)
GZOH (ejω )
0.1
G(jω)
G(jω)e−jωT /2
Magnitude
0.01
π/T
1
10
0
log ω
(rad/sec)
−90
G(jω)
−180
GZOH (ejω )
G(jω)e−jωT /2
Phase (deg.)
2016-5-24
13.5
Digital control system design
Sampled-data closed-loop
y(k)
y(t)
T
G(s)
u(t)
ZOH
u(k)
Kd (z)
+
−
r(k)
K(s) approximation
≈ K(s)
y(k)
T
G(s)
ZOH
Kd (z)
+
r(k)
−
2016-5-24
13.6
Design approaches
G(s)
Continuous-time
design
K(s)
Approximation
of K(s) with
Kd (z)
ZOH-equivalence
of G(s) and
sample/hold
GZOH (z)
Discrete-time
design
Kd (z)
2016-5-24
13.7
Design approaches
G(s)
Continuous-time
design
K(s)
Sampled-data
design
Approximation
of K(s) with
Kd (z)
ZOH-equivalence
of G(s) and
sample/hold
GZOH (z)
2016-5-24
Discrete-time
design
Kd (z)
13.8
Design by approximation
1. Design a continuous-time controller, K(s)
I
Verify stability, performance and bandwidth
I
Verify margins and robustness
2. Select a sample-rate, T
3. Find Kd (z) approximating K(s)
4. Calculate the ZOH-equivalent GZOH (z)
5. Check the stability of the GZOH (z), Kd (z) loop
6. Simulate Kd (z) with G(s) (including sample/hold).
I
Verify simulated performance
I
Examine intersample behaviour
2016-5-24
13.9
Controller approximation
Approach: approximating the integrators
y(t)
x(t)
1/s
y(k)
≈
If F (z) ≈ 1/s, then, s ≈ F −1 (z),
y(s)
2016-5-24
F (z)
x(s)
A
C
B
D
x(k)
=⇒ Kd (z) = K(s) |s=F −1 (z)
1
I
s
sx(s)
F (z)
u(s)
zx(z)
y(z)
x(z)
A
C
B
D
u(z)
13.10
Integration
x(t)
y(t)
x(t)
1/s
y(t) = y(0) +
Z
x(kT +T )
t
x(τ ) dτ,
0
x(kT )
y(kT +T ) − y(kT )
kT
kT +T
Time
[sec.]
The signal, y(t), over a single T second sample period is,
Z kT +T
y(kT + T ) = y(kT ) +
x(τ ) dτ.
kT
2016-5-24
13.11
Trapezoidal approximation
ŷ(kT + T ) = ŷ(kT ) + T x(kT ) + (x(kT + T ) − x(kT ))T /2.
Taking z-transforms,
z ŷ(z) = ŷ(z) + T x(z) +
Approximation:
ŷ(z)
T z+1
= F (z) =
.
x(z)
2 z−1
T
(z − 1)x(z),
2
x(t)
x(kT +T )
So the substitution is,
s ←−
2 z−1
.
T z+1
x(kT )
ŷ(kT +T ) − ŷ(kT )
kT
kT +T
Time
[sec.]
This is known as a bilinear (or Tustin) transform.
2016-5-24
13.12
Frequency mapping
Pole locations under bilinear transform:
{ s | real(s) < 0 }
K(s) stable
bilinear
−→
⇐⇒
{ z | |z| < 1 }
Kd (z) stable.
imag
imag
1
2π/T
π/T
s=
2 (z−1)
T (z+1)
−→
real
real
−1
1
−π/T
−1
−2π/T
2016-5-24
13.13
Bilinear frequency distortion
Ω : discrete-frequencies: ejΩT , Ω ∈ (−π, π].
Frequency mapping:
Continuous frequencies, ω to discrete frequencies, Ω.
Substitute s = jω and z = ejΩT into s =
2 z−1
:
T z+1
−jΩT
2 j sin(ΩT /2)
2
2 1−e
jω =
=
= j tan(ΩT /2).
−jΩT
T (1 + e
)
T cos(ΩT /2)
T
Frequency distortion:
Ω=
2016-5-24
2
tan−1 (ωT /2)
T
13.14
Bilinear frequency distortion
Ω=
2
tan−1 (ωT /2)
T
Tustin/bilinear transform
π
Discrete frequency (Ω): [rad/sec]
Continuous frequency (ω): [rad/sec]
−3π/T
0
−π/T
−2π/T
π/T
2π/T
3π/T
−π
Ω = ωT
The Ω = ωT line is the sampling mapping.
2016-5-24
13.15
Prewarping
s=
α(z − 1)
, α ∈ (0, π/T ), maps {real{s} < 0} to {|z| < 1}.
(z + 1)
Modifying the frequency distortion
Select a frequency ωpw .
Solve for α such that K(jωpw ) = Kd ejωpw T .
The “prewarped” transform makes K(jω) = Kd (ejωT ) at ω = 0 and ω = ωpw .
jω0
=
α(ejωpw T − 1)
(ejωpw T + 1)
which implies that:
2016-5-24
α=
=
jα tan(ωpw T /2),
ωpw
.
tan(ωpw T /2)
13.16
Prewarping
Frequency distortion (bilinear:) Ω =
2
tan−1 (ωT /2).
T
Frequency distortion (with prewarping): Ω =
2
tan−1 (ω/α)
T
Prewarping distortion
Discrete frequency: [rad/sec] 50
Bilinear distortion
Prewarping
frequency
-150
-100
0
-50
50
100
150
Continuous frequency: [rad/sec]
-50
Ω = ωT
2016-5-24
13.17
Controller approximations
Bilinear approximation: Kbl (ejω ), Prewarped Tustin approximation: Kpw (ejω )
10
Magnitude
10
Kbl (ejω )
ωpw
K(jω)
log ω (rad/sec)
1
Kpw (ejω )
0.1
0.01
90
10
0
Kbl (ejω )
log ω (rad/sec)
ωpw
Kpw (ejω )
−90
−180
2016-5-24
K(jω)
Phase (deg.)
13.18
Choosing a prewarping frequency
The prewarping frequency must be in the range: 0 < ωpw < π/T .
I
α = 2/T (standard bilinear) corresponds to ωpw = 0.
Possible options for ωpw depend on the problem:
I
The cross-over frequency (helps preserve the phase margin);
I
The frequency of a critical notch;
I
The frequency of a critical oscillatory mode.
The best choice depends on the most important features in your control design.
Remember: K(s) stable implies Kd (z) stable.
But you must check that (1 + GZOH (z)Kd (z))−1 is stable!
2016-5-24
13.19
Example
Plant model:
G(s) =
2
5(1 − s/zrhp ) (s2 + 2ζηωm s + η 2 ωm
) 1
2
2
(1 + τ s)
(s + 2ζωm s + ωm ) η 2
where
τ = 0.5,
zrhp = 70,
ωm = 20,
ζ = 0.05,
and
η = 1.2.
IMC desgn
Tideal (s) = 3rd order Butterworth filter with bandwidth: 25 [rad./sec.]
Q(s) = Tideal (s)G−1
mp (s)
K(s) = (I − Q(s)G(s))−1 Q(s).
2016-5-24
13.20
Loopshapes
100
Magnitude
L(jω)
10
G(jω)
1
10
K(jω)
100
1
10
100
log ω (rad/sec)
1
0.1
0.01
0.001
90
log ω (rad/sec)
0
K(jω)
−90
L(jω)
G(jω)
−180
−270
Phase (deg.)
2016-5-24
13.21
Sensitivity and complementary sensitivity
10
Magnitude
2
S(jω)
10
100
1
log ω (rad/sec)
T (jω)
0.1
2016-5-24
13.22
Step response
Output
1.4
Amplitude
1.2
1.0
0.8
0.6
0.4
y(t)
0.2
time (sec)
0
0.5
1.0
1.5
2.0
−0.2
2016-5-24
13.23
Step response
Actuation
1.4
Amplitude
1.2
u(t)
1.0
0.8
0.6
0.4
0.2
time (sec)
0
0.5
1.0
1.5
2.0
−0.2
2016-5-24
13.24
Bilinear/Trapezoidal/Tustin transform
Bilinear transform
Nyquist frequency:
Kbl (z) = K(s)
100 radians/second
=⇒
T =
π
.
100
2 z−1
s= T
z+1
Discrete-time analysis
GZOH (z)
y(k)
G(s)
T
Kd (z)
ZOH
+
r(k)
−
2016-5-24
13.25
Loopshapes: bilinearly transformed controller
100
Magnitude
10
1
1
10
100
Kd (ejω )
0.1
log ω (rad/sec)
G(jω)
GZOH (ejω )
0.01
Ld (ejω )
K(jω)
0.001
L(jω)
90
1
10
100
log ω (rad/sec)
0
−90
G(jω)
−180
−270
2016-5-24
Phase (deg.)
K(jω)
GZOH (ejω )
Ld (ejω )
L(jω)
Kd (ejω )
13.26
Sensitivity and complementary sensitivity: bilinearly transformed controller
Magnitude
10
2
S(jω)
1
10
Sd (ejω )
Td (ejω )
100
log ω (rad/sec)
T (jω)
0.1
2016-5-24
13.27
Step response: bilinearly transformed controller
Output
1.4
Amplitude
y(k)
1.2
1.0
0.8
0.6
y(t)
0.4
0.2
time (sec)
0
0.5
1.0
1.5
2.0
−0.2
2016-5-24
13.28
Step response: bilinearly transformed controller
Actuation
1.4
Amplitude
1.2
u(t)
1.0
0.8
u(k)
0.6
0.4
0.2
time (sec)
0
0.5
1.0
1.5
2.0
−0.2
2016-5-24
13.29
Prewarped Tustin transform
Prewarped Tustin transform
Nyquist frequency:
100 radians/second
=⇒
T =
π
.
100
Select the prewarping frequency at ωpw = ωm (20 radians/sec.).
Kpw (z) = K(s)
z−1
s=α z+1
where,
α=
2016-5-24
ωpw
.
tan(ωpw T /2)
13.30
Loopshapes: prewarped Tustin controller
100
Magnitude
10
1
1
10
100
log ω (rad/sec)
Kd (ejω )
0.1
G(jω)
GZOH (ejω )
Ld (ejω )
0.01
K(jω)
0.001
L(jω)
90
1
10
100
log ω (rad/sec)
0
−90
G(jω)
−180
−270
Phase (deg.)
K(jω)
GZOH (ejω )
Ld (ejω )
L(jω)
Kd (ejω )
2016-5-24
13.31
Sensitivity and complementary sensitivity: prewarped Tustin controller
10
Magnitude
2
S(jω)
log ω (rad/sec)
1
10
Sd (ejω )
100
Td (ejω )
T (jω)
0.1
2016-5-24
13.32
Step response: prewarped Tustin controller
Output
1.4
Amplitude
y(k)
1.2
1.0
0.8
0.6
y(t)
0.4
0.2
time (sec)
0
0.5
1.0
1.5
2.0
−0.2
2016-5-24
13.33
Step response: prewarped Tustin controller
Actuation
1.4
Amplitude
1.2
u(t)
1.0
0.8
u(k)
0.6
0.4
0.2
time (sec)
0
0.5
1.0
1.5
2.0
−0.2
2016-5-24
13.34
Sample rate selection
Sample rate selection is critical to digital control design.
Main considerations
I
Sampling/ZOH will (approximately) introduce a delay of T /2 seconds.
I
Anti-aliasing filters will need to be designed and these will also introduce
phase lag.
I
The system runs “open-loop” between samples.
I
Very fast sampling can introduce additional noise.
I
Very fast sampling makes all of the poles appear close to 1. The controller
design can become numerically sensitive.
2016-5-24
13.35
Designing for digital implementation
Sampled-data implementation
d(t)
Gd (s)
u(t)
y(t)
n(t)
+
G(s)
ZOH
ym (t)
+
Fa (s)
u(k)
r(k)
Kd (z)
ỹm (t)
ỹm (k)
T
Continuous-time design
d(t)
Gd (s)
n(t)
2016-5-24
r(t)
u(t)
y(t)
+
+
G(s)
ym (t)
e−sT /2
Fa (s)
K(s)
ỹm (t)
13.36
Sensitivity function
We want a similar discrete sensitivity function up to the frequency where
|S(jω)| returns to 1.
5
Magnitude
1
10
+
1
log ω
(rad/sec)
ωB
−
0.1
S(jω)
0.01
S(s) = (I + Fa (s)G(s)e−sT /2 K(s))−1
2016-5-24
13.37
Sensitivity function
In this example, for ω > 20 rad./sec.,
|1 − S(jω)| 1
5
=⇒
π/T = 20
is about the minimum.
Magnitude
10
+
1
−
S(jω)
20
ω
(rad/sec)
ωB
0.1
0.01
S(s) = (I + Fa (s)G(s)e−sT /2 K(s))−1
2016-5-24
13.38
Loop-shaping interpretation
For ω up to where |Fa (jω)G(jω)K(jω)| < and remains very small,
jωT
−jωT /2
jωT
.
we want Fa (jω)G(jω)K(jω)e
≈ ĜZOH e
Kd e
(ĜZOH (z) is the ZOH-equivalent of Fa (s)G(s))
imag
j
|S(jω)| > 1
−1
1
real
|1 + L(jω)|
L(jω)
−j
2016-5-24
13.39
Fast sampling
Fast sampling period: Tf .
Control appropriate (slower) sampling period: Ts
(typically Ts = M Tf for integer M > 1).
ỹs (k)
Ts
Fas (z)
Digital
Digital
downsampling filter
2016-5-24
ỹ(t)
Tf
Fast
sampler
Faf (s)
y(t)
Anti-aliasing
filter
13.40