Motivation PID Controller Design Feedforward Control Conclusion & Outlook PID Controller Design for Nonlinear Systems Using Discrete-Time Local Model Networks 4. Workshop f¨ ur Modellbasierte Kalibriermethoden Nikolaus Euler-Rolle, Christoph Hametner, Stefan Jakubek Christian Mayr (AVL List GmbH) 08.11.2013 4. Workshop f¨ur ModellbasierteKalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 1/26
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Motivation PID Controller Design Feedforward Control Conclusion & Outlook
PID Controller Design for Nonlinear Systems UsingDiscrete-Time Local Model Networks
4. Workshop fur Modellbasierte Kalibriermethoden
Nikolaus Euler-Rolle, Christoph Hametner, Stefan Jakubek
Christian Mayr (AVL List GmbH)
08.11.2013
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 1/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Feedback Control of Nonlinear Systems
Motivation
Implementation of Two-Degrees-of-Freedom control using local model networks
Feedforward part improves the dynamic performance- Reference tracking- Deadtime- Input saturation
Controller design on (semi)-physical process models instead of testbed runs
Opportunity of inexpensive feasibility studies and rapid prototyping
PID Plant
ww*
u*
uy
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 2/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Feedback Control of Nonlinear Systems
Motivation
Implementation of Two-Degrees-of-Freedom control using local model networks
Feedforward part improves the dynamic performance- Reference tracking- Deadtime- Input saturation
Controller design on (semi)-physical process models instead of testbed runs
Opportunity of inexpensive feasibility studies and rapid prototyping
Approach
Globally nonlinear process model (based on input/output measurements)
Design of nonlinear PID controllers with guaranteed global stability
Fully automated generation of a dynamic feedforward control
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 2/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Controller Design Workflow
TestbedMaps
Signals
DoE
[n, q, u]
LMNSS-ModelLocal PIDs
DoE
Optimisation
y
SimulationParameter
ControllerMaps
Performance
Stability
Ide
ntifica
tio
n
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 3/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Controller Design Workflow
TestbedMaps
Signals
DoE
[n, q, u]
LMNSS-ModelLocal PIDs
DoE
Optimisation
y
SimulationParameter
ControllerMaps
Performance
Stability
Ide
ntifica
tio
n
DynamicFF-Control
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 3/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
2 PID Controller Design
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 4/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Local Model NetworkOverview
1000 1500 2000 2500
5
10
15
20
25
30
InjectionMass,mg/stroke
Engine Speed, rpm
local
global
Local Model Network
Globally nonlinear dynamical systemrepresented by local linear models
Found by system identification
Local stability proof & controllerdesign using linear methods
⇒ Global approach necessary (due totransition, model interpolation...)
o for nonlinear systemso based on Lyapunov stability theory
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 5/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Typical PID Controller StructureExample: Engine Control Unit
-
min
max
P-Part
I-Part
DT1-Part
anti windup
Map
Map
Feedforward-
Feedback-Control
n
n
n
q
q
q
u
w
y
e
ufb
uff
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 6/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Feedback Controlled Local Model Network
Concept
One local controller (LC) per local model(LM)
Scheduling of parameters according to thevalidity functions of local models (ParallelDistributed Compensator)
KPID(Φ) =∑
ΦiK(i)PID
Formal split into inputs used for control uand disturbances z
!"#$%"&&'%
(')*+#
,-!.
,- /
,!!.
,! /+
01'%2$*#+!
1"*#$!
('1'#('#$
3,-!/
,--
,!!/
,! -,-!- ,!!-
Nonlinear process is approximated by a local model networkTrade-Off: model fit ↔ simple controller design
Consider an undamped oscillator with anonlinear spring force characteristicf(y), which is to be stabilized usingconstant c and input u
y + f(y) = cu
Figure: Air suspension
Exact Linearization
For this second order system, the state variables are chosen as
y = x1
y = x1 = x2
y = x1 = x2 = cu− f(y) = v
1
s
1
s
v y
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 17/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Feedforward ControlUndamped Nonlinear Oscillation
Exact Linearization
For a two times differentiable desired trajectory w, the nonlinear feedforward controlinput u∗ can be found from
v!= w = cu∗
− f(w) → u∗ =w + f(w)
c
1
s
1
s
yu
u∗
w
w
C
u∗ =w + f(w)
c
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 18/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Demonstration ExampleAutomatic Feedforward Control Design
Wiener Model
G(z) =P (z)
U(z)=
0.6z−3
1− 1.3z−1 + 0.8825z−2 − 0.1325z−3
y(k) = f(p(k)) = arctan(p(k))
Figure: Wiener Model approximated by an LMN:
y(k
−1)
u(k − 3)
6
5
43
2
1
−3 −2 −1 0 1 2 3
−1
−0.5
0
0.5
1
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 19/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Feedforward Controlled SimulationWiener Model
Samples
yW
iener
uw,y
40 60 80 100 120 140 160 180 200 220
40 60 80 100 120 140 160 180 200 220
40 60 80 100 120 140 160 180 200 220
−1
0
1
−3
0
3
−1
0
1
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 20/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Feedforward Controlled SimulationWiener Model
y FFCw
y
Samples0 50 100 150 200 250 300 350 400 450 500
−1.5
−1
−0.5
0
0.5
1
1.5
PID Plant
ww*
u*
uy
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 21/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Two-Degrees-of-Freedom ControlWiener Model
y 2DoFy FFCw
Samples
y
0 50 100 150 200 250 300 350 400 450 500−1.5
−1
−0.5
0
0.5
1
1.5
PID Plant
ww*
u*
uy
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 22/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Two-Degrees-of-Freedom ControlWiener Model
y 2DoFyPIDw
Samples
y
0 50 100 150 200 250 300 350 400 450 500−1.5
−1
−0.5
0
0.5
1
1.5
PID Plant
ww*
u*
uy
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 23/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
4 Conclusion & Outlook
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 24/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Conclusion & Outlook
Two-Degrees-of-Freedom Control
Nonlinear PID controller design using local model networks
Multi-objective optimisation of controller parameters consideringStabilityPerformance
Automatic feedforward control law generation for minimum-phase local modelnetworks
Outlook
Application of a Lyapunov function to check internal stability
Considering input constraints
Assessment of Two-Degrees-of-Freedom control on a physical process
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 25/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Thank you for your attention!
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 26/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Fitness Function: Stability
Common Quadratic Lyapunov Function for Closed-Loop Systems
Exponential stability with decay rate α of the closed-loop feedback system is shown, ifsymmetric matrices P and Xij exist and the following conditions are fulfilled:
P ≻ 0
inf{
0 < α < 1 : ΛTijPΛij + Xij � α
2P}
X =
X11 X12 · · · X1I
X12 X22 · · · X2I
.
.
.. . .
.
.
.X1I X2I · · · XII
≻ 0
∀i ∈ I, ∀i ≤ j ≤ I
using
Λij =Gij + Gji
2,
Gij = Ai − BikTPID,jC.
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 27/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Fitness Function: Stability
Common Quadratic Lyapunov Function for Closed-Loop Systems
Exponential stability with decay rate α of the closed-loop feedback system is shown, ifsymmetric matrices P and Xij exist and the following conditions are fulfilled:
P ≻ 0
inf{
0 < α < 1 : ΛTijPΛij + Xij � α
2P}
X =
X11 X12 · · · X1I
X12 X22 · · · X2I
.
.
.. . .
.
.
.X1I X2I · · · XII
≻ 0
∀i ∈ I, ∀i ≤ j ≤ I
using
Λij =Gij + Gji
2,
Gij = Ai − BikTPID,jC.
Simultaneous solving for P and kTPID,j is not possible! → fS = α
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 27/26