Post on 29-Dec-2021
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
Closed-loop state-space representation necessary (to prove Lyapunov stability)
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 7/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Closed-Loop State-Space RepresentationIncluding Error Signal Adaptation
InputScheduler
q−1I
v(k)B(Φ)
B(Φ)
E(Φ) A(Φ)
KPID(Φ, e)
we(Φ, e)
cTx(k + 1)
x(k)
y(k)w(k)
z(k)z(k)
f(Φ)
-
SystemPre-Filter
Figure: Local model network with PID controller in state-space representation
State Equation
x(k + 1) = [A(Φ)−B(Φ)KPID(Φ, e)]x(k) +B(Φ)G(Φ, e)w(k) +E(Φ)z(k)
+ f(Φ) +B(Φ)we(Φ, e)
y(k) =cTx(k)
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 8/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Overview of the Design Procedure
Controller Design
Basic calibration (linear design methods per local model)
Generation of a suitable performance sequence (DoE)- Operating range (e.g.: 1000–4000 rpm, 0–70 mg/stroke)- Holding time- Gradients (e.g.: engine speed)- Filtering
Nonlinear, multi-objective optimisation of controller parameters considering- Performance- Stability
Multi-objective optimisation of the parameters of the error signal adaptation(optional)
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 9/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Multi-Objective Genetic Algorithm
Objective Function
min fm(xopt)subject to gj(xopt) ≥ 0
hk(xopt) = 0
x(lb)i ≤ xi ≤ x
(ub)i
fS Stability (by Lyapunov’s direct method)
fP Performance (by a closed-loop simulation)
0
fP
fS
Paretofrontier
GA Population
1 n
· · ·
· · ·Individuals
Gen
ome
Gen
ome
Fitness
Fitness
Stability
Stability
Perform
ance
Perform
ance
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 10/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Fitness Function: StabilityLyapunov’s Direct Method for Discrete-Time Systems
Stability of Dynamic Systems
A positive definite, scalar Lyapunov functionV (k) = V (x(k)) with state vector x(k)proves global asymptotic stability if:
o V (x(k) = 0) = 0
o V (k) > 0 for x(k) 6= 0
o V (k) → ∞ as ‖x(k)‖ → ∞
o V (k + 1) < V (k) ∀k ∈ N+
or global exponential stability if:o V (k + 1) ≤ α2V (k) ∀k ∈ N
+
with decay rate 0 < α < 1
Results in Linear Matrix Inequalities (LMIs),which are solved by optimisation
Sufficient but not necessary condition
Common Quadratic Lyapunov Function
V (k) = xT(k)Px(k)
LMI Problem
P ≻ 0
inf{
0 < α < 1 :
ΛTijPΛij +Xij � α2P
}
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 11/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Fitness Function: Performance
Requirements
Assessment of the closed-loopperformance for a given set ofparameters
Representative synthetic reference isgenerated by DoE
Desired trajectory is PT1-filtered
Fitness Function
Closed-loop simulation of the referencecycle for each genome
Sum of squared errorsfP =
∑
∀k(y(k)− ydmd(k))2
Time
0 5 10 150
0.5
1
1.5
2
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 12/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Pareto-Optimal Solutions
0.995 1 1.005 1.01 1.015 1.02 1.025 1.03
1
1.5
2
2.5
3
3.5
4x 10
7
A
B
Pe
rfo
rma
nce
Stability
fS
fP
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 13/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
3 Feedforward Control
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 14/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Feedforward Control
State of the Art: Static Model Inversion
Steady state input is found by static model inversion
u(Φ) = [cT(I − A(Φ))−1B(Φ)]−1(w(Φ) − c
T(I − A(Φ))−1(E(Φ)z(Φ) + f(Φ)))
Stored in a map
Dynamic Feedforward Control
PID Plant
ww*
u*
uy
Dynamic feedforward control improves the closed-loop performance.
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 15/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Dynamic Feedforward ControlGeneration using Local Model Networks
Benefits
Automatic generation of a dynamic feedforward control law for nonlineardynamic systems
Exploits the generic model structure of local model networks
Model complexity may be arbitrarily high
Applicable online for any reference trajectory without pre-planning
Properties
Based on an open-loop state-space model
Realised by a feedback linearizing input transformation
Restricted to globally minimum-phase local model networks
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 16/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Feedback LinearizationUndamped Nonlinear Oscillation
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)
4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 17/26
Motivation PID Controller Design Feedforward Control Conclusion & Outlook
Feedback LinearizationUndamped Nonlinear Oscillation
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